Finding Gaussian Structure in Bosonic States

Finding Gaussian Structure in Bosonic States Alvan Arulandu∗

Sitan Chen†

Ziyun Chen‡

Jerry Li§

Eric Ma¶

arXiv:2610.06810v1 [quant-ph] 5 Oct 2026

October 6, 2026

Abstract We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary 𝑛-mode bosonic state 𝜌, the goal is to output a pure Gaussian state whose infidelity with 𝜌 is at most opt + 𝜖, where opt is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When opt is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in 𝑛, 1/𝜖 and log log 𝐸, where 𝐸 is the energy of the closest pure Gaussian state. For arbitrary opt, our protocol uses (𝑛 + 1) poly(1/𝜖 ) poly (1 + log log(𝐸)) copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether opt > 𝑐 + 𝜖 or opt < 𝑐 − 𝜖, for any threshold 𝑐 ∈ (0, 1). We also prove poly(𝑛, 1/𝜖) runtime is impossible, unless NP ⊆ BQP. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

∗ Harvard University, aarulandu@college.harvard.edu. † Harvard University, sitan@seas.harvard.edu

‡ University of Washington, ziyuncc@cs.washington.edu § University of Washington, jerryzli@cs.washington.edu ¶ Stanford University, eryma@stanford.edu

Contents 1

Introduction 1.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Our Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 2 3 5

2

Preliminaries 2.1 Matrix decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bosonic quantum systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Primitives from robust statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 8 13

3

Agnostic learning in the high-fidelity regime 3.1 Robust heterodyne tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Robust iterative unsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Robust pure state projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Robust iterative refinement via gradient descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Near-vacuum geometry of the infidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 Noisy objective estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.3 Gradient descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Direct gradient estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14 14 15 16 17 18 22 22 24

4

Agnostic learning in the low-fidelity regime 4.1 List-decodable iterative unsqueezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 List estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Unsqueezing and the fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Pruning the list . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 The algorithm and its analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Optimization at bounded photon number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Learning the low degree components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Global optimization of the normalized overlap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25 25 26 27 28 33 37 37 39 41 45

5

Agnostic learning with Gaussian measurements 5.1 Refining to 𝑂 (opt) + 𝜖 with Gaussian measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hardness for Gaussian measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47 47 50

6

Hardness in the low-fidelity regime

54

A Deferred proofs A.1 Deferred proofs from Section 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Proof of Lemma 2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.2 Proof of Lemma 2.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Deferred proofs from Section 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.1 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.2 Proof of Theorem 3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.3 Proof of Lemma 3.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.4 Proof of Lemma 3.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2.5 Proof of Lemma 3.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3 Deferred proofs from Section 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.3 Proof of Lemma 4.10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.4 Proof of Lemma 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2

65 65 65 66 67 67 68 68 69 71 72 72 73 73 74

A.4 Deferred proofs from Section 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4.1 Proof of Lemma 5.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Deferred proofs from Section 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5.1 Proof of Lemma 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

75 75 75 75

B Tensor representation of the truncated numerator

78

C Subgradient and projection oracles

80

1

Introduction

Gaussian states are the backbone of bosonic quantum information. They are the states generated by the workhorse operations of quantum optics such as displacement, squeezing, and passive linear optics, and they are what experiments prepare when they inject squeezed light into gravitational-wave detectors [GDD+ 13, TYK+ 19, AAA+ 19, GJN+ 23] and into microwave haloscopes searching for axion dark matter [MPB+ 19, BPK+ 21], or when they send squeezed light through a large interferometer to perform Gaussian boson sampling [HKS+ 17, ZWD+ 20, MLA+ 22]. Gaussian states also play the role of mean-field theory for bosons: just as product states and Slater determinants are the mean-field ansätze for qubits and fermions, Gaussian states are the variational family behind Bogoliubov theory and its modern descendants [Bog47, SDC18, HGS+ 20]. Yet, the bosonic states that matter most for quantum technology are typically not Gaussian. However, many important states that arise both in physical experiments and in the design of quantum computers are often close to Gaussian. This can be due to noise during state preparation: for instance, fluctuations of the relative phase between a squeezer and its reference turn a squeezed state into a mixture of rotated squeezed states [FHD+ 06, DBC+ 13, KMI+ 14], and in a superconducting cavity the self-Kerr nonlinearity inherited from a dispersively coupled transmon continually bends displaced states away from Gaussianity [KVL+ 13]. Alternatively, when Gaussian states are used in mean-field approximations, the real underlying state is understood to be only well-approximable by Gaussian states. Indeed, non-Gaussianity is a crucial concept in bosonic quantum information, underlying essentially every route to bosonic quantum advantage and fault tolerance, from cat qubits [VKL+ 13, LVP+ 20, GFP+ 20, PNH+ 25] and Gottesman-Kitaev-Preskill (GKP) codes [GKP01, FNM+ 19, CIET+ 20, SER+ 23b, KAH+ 24, LBK+ 25] to modular photonic quantum computers [AAA+ 25]; see e.g. [Wal21, LGO+ 20] for reviews. In both situations, it is natural to ask what Gaussian structure the state contains, and how much of the state this structure accounts for. To this end, we study the task of agnostic tomography of pure bosonic Gaussian states. We consider the following problem: given copies of an 𝑛-mode bosonic state 𝜌, we would like to find a classical description of a pure Gaussian state |𝜓⟩ maximizing the fidelity 𝐹 (𝜌, 𝜓) = ⟨𝜓| 𝜌 |𝜓⟩. More formally, if we let G denote the set of pure Gaussian states, for a given state 𝜌, we can let opt = 1 − sup ⟨𝜓| 𝜌 |𝜓⟩

(1)

| 𝜓⟩ ∈ G

denote the best infidelity to 𝜌 achievable by any Gaussian state. A (proper) agnostic tomography algorithm ˆ ∈ G so that 1 − ⟨𝜓| ˆ 𝜌 | 𝜓⟩ ˆ ≤ opt + 𝜖. with error parameter 𝜖 > 0 is one which, given copies of 𝜌, outputs a | 𝜓⟩ Our goal is to achieve such an algorithm that is efficient, both in terms of the number of copies of 𝜌 it requires, as well as in its overall runtime. An algorithm for this task would have a number of natural applications. First, it would immediately imply ˆ This is a natural an algorithm for estimating opt to error 𝜖, by simply measuring the overlap of 𝜌 with | 𝜓⟩. fidelity-based measure of non-Gaussianity, in the spirit of resource theories of non-Gaussianity [TZ18, AGPF18]: it is the bosonic analogue of the stabilizer fidelity [CGYZ25] and of the geometric measure of entanglement [BBK+ 25]. It also differs from the distance-based measures of [GPB07, GPB08, GMM13] which compare 𝜌 the Gaussian state with the same first and second moments, rather than against all possible Gaussian states. Estimating it would allow us to measure the utility of mean-field approximations in realworld settings. The same quantity sets the threshold in fidelity-based witnesses of stellar rank [CMG20a,

1

CRW+ 21, FNB+ 24]: in those works, opt is explicitly computed classically for a known non-Gaussian target state known target state, and any experimental state whose fidelity with the target exceeds it is thus certified to be non-Gaussian. In contrast, our algorithm estimates the Gaussian fidelity of the unknown state directly, with no reference state. But agnostic tomography would allow us to do much more than just estimating the fidelity. Writing ˆ = 𝐺 |0⟩ for a Gaussian unitary 𝐺, it singles out the rotated Fock basis {𝐺 |𝑚⟩}, in which 𝜌 is as close | 𝜓⟩ to the vacuum as possible. For a state which is close to a Gaussian state, this is the state that the device is effectively trying to prepare, and thus serves as a natural Gaussian “sanitization” of the state. For more general states, one can interpret this basis as a Gaussian skeleton: in this basis, most of the remaining non-Gaussian structure sits at low photon number, where it is most economical to describe, in the spirit of the stellar representation of non-Gaussian states [CMG20a, FNB+ 24].

1.1

Results

We give the first protocols for agnostic tomography of pure bosonic Gaussian states in (in)fidelity in both the high-fidelity and low-fidelity regimes. 𝑑IF (𝜌, 𝜎) ≡ 1 − 𝐹 (𝜌, 𝜎) denote the infidelity between two √︁√ Let √ 2 quantum states, where 𝐹 (𝜌, 𝜎) ≡ (tr 𝜌𝜎 𝜌) is the fidelity. Further, let G denote the class of pure bosonic Gaussian states. Informally, our results are as follows: Theorem 1.1 (Agnostic tomography in high-fidelity regime (informal, see Theorem 3.8)). There is a universal constant opt ∈ (0, 1) such that the following holds. Suppose 𝜌 is an arbitrary 𝑛-mode bosonic state, and assume that opt ≤ opt, where opt is defined in Eq. (1). Let 𝐸 be the energy of some closest state in G. Then, there exists an algorithm which takes an 𝜖 > 0 along with poly(𝑛, 𝜖 −1 , log log 𝐸) copies and, using only ˆ ∈ G satisfying Gaussian unitaries, heterodyne measurements, and photon counting, outputs a | 𝜓⟩ ˆ ≤ opt + 𝜖 𝑑IF (𝜌, | 𝜓⟩) with probability at least 0.99. To our knowledge, this is the first robust Gaussian state learning guarantee for any value of opt which is bounded away from 0. Additionally, whereas the above guarantee applies in the “high-fidelity” regime where the unknown state is sufficiently close to Gaussian, we also devise an algorithm in the “low-fidelity” regime where the unknown state is arbitrary: Theorem 1.2 (Agnostic tomography in low-fidelity regime (informal, see Theorem 4.19)). Suppose 𝜌 is an arbitrary 𝑛-mode bosonic state. Let 𝐸 be the energy of some closest state in G. Then, there exists an algorithm which takes an 𝜖 > 0 along with poly(𝑛1/𝜖 , 𝜖 −1 , log log 𝐸) copies and, using only Gaussian ˆ ∈ G satisfying unitaries, heterodyne measurements, and photon counting, outputs a | 𝜓⟩ ˆ ≤ opt + 𝜖 𝑑IF (𝜌, | 𝜓⟩) with probability at least 0.99. Thus, for any arbitrarily small constant fidelity, we can approximate the closest Gaussian state in time and samples polynomial in the system size. As an immediate corollary, our protocol yields the first tolerant tester for Gaussianity that can distinguish between states for which opt > 𝑐 +𝜖 versus opt < 𝑐 −𝜖 for any threshold 𝑐. 2

This is the first guarantee of its kind for bosonic states. Recent algorithms for Gaussianity testing [GWM+ 25, IOP+ 26, YPH+ 26] only tolerate small constant-infidelity deviations from Gaussianity, enabling one to detect when a bosonic state is either close or far from being Gaussian, but not to distinguish between fidelity above or below a specified threshold, even if that threshold is, say, merely 1/2. In other words, ours is the first guarantee for bosonic states, whether for learning or for testing, in the low-fidelity regime, where the best Gaussian fidelity is an arbitrary constant bounded away from one. Analogous low-fidelity guarantees were previously known only for discrete variable systems [GIKL24b, CGYZ25, ADGG26, BBK+ 25, BHN26]. Finally, we pause to remark upon two notable features of our results. First, the guarantee of Theorem 1.2 scales exponentially in the target precision 𝜖. While one might hope for a poly(𝑛, 𝜖 −1 ) bound, in Section 6 we prove that this is impossible: Theorem 1.3 (Fully polynomial runtime is impossible (informal, see Theorem 1.2)). Agnostic tomography of pure Gaussian states is at least as hard as finding maximal cliques in graphs. Thus, unless NP ⊆ BQP, there is no algorithm for agnostic tomography of pure Gaussian states which achieves runtime poly(𝑛, 1/𝜖). In addition, our protocols crucially rely on non-Gaussian measurements, namely photon counting. In Section 5, we show that this is in fact necessary, giving an intriguing new separation between Gaussian measurements and non-Gaussian measurements: Theorem 1.4 (Gaussian measurements are provably weaker than non-Gaussian (informal, see Theorems 5.1 and 5.6)). In the high-fidelity regime, there is a time-efficient protocol for agnostic tomography of pure Gaussian states which only uses Gaussian measurements and achieves infidelity 𝑂 (opt) + 𝜖 using 𝑂˜ (𝑛2 /𝜖) copies. However, there exist settings of opt such that any protocol in this regime that only uses Gaussian
measurements cannot achieve infidelity opt + 𝜖 without using exp Ω(𝑛1/3 ) copies.

1.2

Our Techniques

Our algorithms arise from a shared paradigm, which we believe may be generalizable to other settings. At its heart, our algorithms follow a two-step procedure, which is then possibly iterated depending on the setting: 1. First, use heterodyne measurements to reduce the problem to an instance of a classical problem from high-dimensional robust statistics, and leverage existing techniques from that literature to obtain a good warm-start estimate of the best Gaussian approximation. 2. Leverage this warm-start to rotate the states so that in this new basis, the best Gaussian approximation approximately lives in a low photon number subspace. Finding the closest Gaussian state then amounts to solving a certain tensor optimization problem in this subspace. We expand upon these two points below. Heterodyne Gaussian measurement and robust statistics Gaussian measurements are a powerful, experimentally feasible, class of measurements one can perform on a bosonic state. Crucially, when performed on a Gaussian state, the output distribution of the measurement is a sample from a classical high-dimensional Gaussian, whose parameters correspond directly to the parameters of the underlying state and the specific Gaussian measurement applied (see Section 2 for formal definitions).

3

A key implication of all of this for our application is that if we apply Gaussian measurements to our (potentially non-Gaussian) state 𝜌, the result of this measurement is a draw from a classical distribution which is close in total variation distance to a Gaussian whose underlying parameters would recover the best bosonic Gaussian state approximation to 𝜌. This is a well-studied problem in the field of robust statistics, and by now, many efficient algorithms are known for this problem in a variety of settings. This is precisely the oblivious contamination model of robust statistics [Hub64, DKK+ 16], and the agnostic problem inherits all of the difficulties of that setting. Fortunately, a long line of work in algorithmic robust statistics, starting with [DKK+ 16, LRV16], has produced computationally efficient estimators for high-dimensional distributions whose error does not grow with the dimension; see [DK19] for a textbook treatment. In particular, given samples from a distribution 𝐷 with total variation distance at most 𝜂 from a Gaussian, this e(𝜂) in total variation distance [DKK+ 16]. Gaussian can be estimated in polynomial time to within 𝑂 There are two key settings that are relevant for our problem. When the infidelity of the optimal solution is low (that is, the state 𝜌 is close to being Gaussian), this becomes the standard problem of robustly learning a Gaussian mentioned above. For this problem, the computational landscape is well understood, and there are by now well-established polynomial time algorithms for this problem with strong error guarantees. It turns out that this is already sufficient to obtain a good warm start for the problem: by iteratively combining robust Gaussian estimators with techniques and bounds from [BMEL25, CMF+ 26], this yields a pure Gaussian e(opt), using only Gaussian measurements and poly(𝑛, log log 𝐸) copies state whose infidelity with 𝜌 is 𝑂 (Theorem 3.4), where 𝐸 is an upper bound on the energy of the best Gaussian state approximation (see Section 2). The low-fidelity regime turns out to be considerably more challenging, since here the fidelity 𝛼 = 1 − opt of 𝜌 with its best Gaussian approximation 𝜋∗ may be small. It is then information-theoretically impossible to output a single good estimate. However, this is the setting of list-decodable learning, introduced in [BBV08, CSV17], which has since yielded efficient list-decodable algorithms for mean estimation [CSV17, DKS18] and, most relevant to us, for estimation of general Gaussians [IK22]. We use the list-decodable Gaussian learner of [IK22] on the heterodyne outcomes to obtain a list, of size depending only on 𝛼, of candidate means and covariances, at least one of which approximates the parameters of 𝜋∗ up to factors depending only on 𝛼. Unlike in the high-fidelity regime, such a coarse estimate is not by itself a good Gaussian approximation, but it is good enough to “unsqueeze” the state, that is, it allows us to find a rotated basis in which the true state has smaller energy. We develop a list-decodable variant of iterative unsqueezing which follows every candidate in the list through the corresponding change of basis, and then prunes the resulting bases using a photon-number-based notion of distance between bases, so that only 𝑂 (1/𝛼) bases survive each round. After 𝑂 (log log 𝐸) rounds, this produces a short list of rotated bases, in at least one of which 𝜋∗ has mean photon number at most poly(1/𝛼), independent of both 𝑛 and 𝐸 (see Section 4.1). While these methods already yield non-trivial guarantees, they unfortunately run into some fundamental bottlenecks. In particular, even in the simpler low-infidelity setting, classical robust statistics techniques can e(opt), as opposed to the strong agnostic only ever hope to achieve “semi-agnostic” guarantees, that is, error 𝑂 guarantee of opt + 𝜖 that we would ideally hope for. It is known that such bottlenecks are inherent in classical robust statistics, at least for large classes of algorithms such as SQ algorithms [DKS17]. In fact, in Section 5, we show that no algorithm using only Gaussian measurements can achieve strong agnostic guarantees for this problem. The situation is even worse in the high infidelity setting: here, if we only used the guarantees known from the robust statistics literature, we would achieve final error guarantees which are almost trivial.

4

Low photon number projections and tensor optimization. To rectify this, we introduce a second algorithmic step which utilizes the prior methodology as a warm start, and uses it to reduce the problem to a optimization problem over a manageable, low-dimensional space. We now explain this step in more detail. Rather than directly interpreting the output of a first step as the final output of the algorithm, we view it as a rotated basis 𝐺 (or, in the low-fidelity regime, a short list of candidate bases) in which the best Gaussian approximation is close to the vacuum state. Therefore, in such a basis, the problem collapses onto a low photon number subspace (see Section 2 for formal definitions). Indeed, any pure Gaussian state with mean photon number at most ℎ places all but an 𝜖-fraction of its mass on Fock states with at
most 𝐾 = 𝑂 (ℎ + 1) 3 + (ℎ + 1) log(1/𝜖) total photons (Lemma 4.12). Consequently, its fidelity with 𝜌 is determined, up to small error, by the projection of 𝜌 onto this subspace, even though 𝜌 itself may have arbitrarily large photon number. This is in the spirit of the stellar representation of non-Gaussian states [CMG20b]: in the right rotated basis, the non-Gaussian structure that matters sits at low photon number. It can also be thought of as a natural bosonic analog of the key structural statement utilized by [BBK+ 25] (see e.g. Lemma 5.17 in that paper). In the high-fidelity regime, we show that the warm-start is sufficiently good so that, in the obtained basis, only the vacuum, one-photon, and two-photon sectors matter. Moreover, we demonstrate that the infidelity is locally smooth and strongly convex in a neighborhood of the vacuum. Therefore, we can directly run gradient descent on 𝑓𝜌 , re-centering the basis at the current iterate after every step so that we only ever need first-order information at the vacuum. This by doing so, we are able to refine the warm start to infidelity e(𝑛2 /𝜖) additional copies. We refer the reader to Section 3.4 and Theorem 3.8 for a full opt + 𝜖 using 𝑂 description and analysis of our algorithm. In the low-fidelity regime, no such local convexity is available, and the optimization is genuinely global. However, our warm start still allows us to project the problem onto the subspace spanned by all Fock states with at most 𝐾 photons, which has dimension (𝑛 + 1) 𝐾 . This allows us to develop the bosonic analog of the tensor optimization procedure used in [BBK+ 25] for outputting a good product state approximation given a sufficiently good warm start estimate. At a high level, we demonstrate how to reduce this problem to an approximate tensor optimization problem on a tensor with bounded Frobenius norm. We can then use the tensor optimization routine developed in [BBK+ 25] to obtain our final solution. Interestingly, our reduction is not black-box: this is because a primary difficulty in the reduction is to deal with the fact that the states must be normalized naively renders the problem into a rational optimization problem, rather than a polynomial optimization problem. In [BBK+ 25], they deal with this problem by simply brute-forcing enough quantities to allow them to ignore this constraint. We cannot do this: instead, we show that this problem can be, in large part, ignored. As in [BBK+ 25], we show that the unnormalized problem can be reduced to an enumeration problem over a small list of possible answers, where each answer corresponds to a set of candidate solutions, projected down to a low-dimensional subspace. We show that the task of associating each projected solution back to normalized Gaussian state with as small norm as possible is in fact a convex optimization problem. Hence, we can efficiently solve this problem, and thus directly compute the answer of the normalized problem for each candidate solution the unnormalized one. We defer the reader to Section 4.2 for a full algorithm description and analysis.

1.3

Related work

Agnostic tomography. Agnostic tomography was proposed in recent work [BO21, AA24, GIKL24a] and has been studied for stabilizer product states [GIKL24a], stabilizer states and states of high stabilizer dimen5

sion [GIKL24b, CGYZ25, ADGG26], product states [BBK+ 25, BHN26], and Slater determinants [PZD26]. [ADKL25] gave the first guarantees for a class of mixed states, namely product mixed states, through a connection to robust learning of binary product distributions. The present work continues this connection between agnostic tomography and robust statistics, establishing the first link to the rich literature on classical robust statistics over Euclidean space (see below). Related notions include agnostic process tomography [WLKD25] and learning from adversarially corrupted copies [ABCL25]. To our knowledge, ours is the first work to study agnostic tomography in the continuous-variable setting. Learning bosonic Gaussian states. The theory of bosonic quantum learning has developed rapidly and we refer to the recent survey of [Mel26] for a comprehensive overview. Mele et al. [MMB+ 25] showed that general bosonic states require a number of copies exponential in the number of modes, while Gaussian states and states with few non-Gaussian gates can be learned efficiently. Subsequent work on Gaussian state learning reduced the energy dependence to log log 𝐸 through adaptive unsqueezing [BMEM25, CMF+ 26], which was subsequently shown to be unavoidable for Gaussian seed measurements [CGYZ26]. The optimal sample complexity of Gaussian state tomography was determined in the pure state case by [CGYZ26] and in the general case by [CFG+ 26]. Recent work also characterized when Gaussian states are as easy to learn as classical Gaussians [CMMP26]. All of these results pertain to the setting where the unknown state is exactly Gaussian, whereas our work is the first to consider the agnostic setting. Testing and quantifying non-Gaussianity. There are multiple different ways of quantifying non-Gaussianity that have been considered in the literature, either through distances to Gaussian reference states [GPB07, GPB08, GMM13], through resource theories [TZ18, AGPF18], through the stellar rank [CMG20a], through the symplectic rank [MOUC25], or through correlations generated at a beam splitter [HT25]. There are a number of proposed ways to certify non-Gaussianity experimentally, either through photon-counting statistics [FM11, JSM+ 11, LSH+ 19], fidelity with specific non-Gaussian target states whose maximal Gaussian fidelity is computed classically [CRW+ 21, FNB+ 24], and fidelities with specific Gaussian targets using Gaussian measurements [AGKE15]. In contrast, our work is the first to show that it is possible to provably estimate the maximum fidelity with respect to all Gaussian states. On the algorithmic side, [GWM+ 25] introduced symmetry-based tests that accept every pure Gaussian state with certainty, showed that they reject states 𝜖-far from Gaussian using a number of copies polynomial in the number of modes and the energy, gave a somewhat tolerant variant, and proved that testing whether a mixed state is Gaussian requires exponentially many copies. Two concurrent works [IOP+ 26, YPH+ 26] give testers for pure states using Θ(1/𝜖 2 ) copies. The resulting testers distinguish between states within trace distance 𝜖 1 of Gaussian from states at trace distance at least 𝜖 2 , but only for restrictive settings of 𝜖1 , 𝜖2 . In [IOP+ 26], they require 𝜖 2 > 𝛽𝜖 1 for large constant 𝛽; in [YPH+ 26] they require 𝜖2 = (1 + 𝜂)𝜖1 with 𝜖 1 ≤ 𝑐𝜂 and use joint measurements on 𝑂 (1/𝜂) copies [YPH+ 26]; in [GWM+ 25] they require 𝜖 1 ≤ 𝐶 min{𝜖22 , (𝑛𝐸) −4 }. In all of these cases, 𝜖 1 must thus be at most a small constant, so that the testers certify fidelity close to one. Our algorithm can instead be used to distinguish opt ≤ 𝑐 − 𝜖 from opt ≥ 𝑐 + 𝜖 for any constant 𝑐 ∈ (0, 1), answering an open question of [GWM+ 25]. Additionally, whereas [IOP+ 26, YPH+ 26] assume pure inputs, our protocol works on arbitrary and possibly mixed inputs. Nevertheless, our guarantee is incomparable as we require copies and time polynomial in the number of modes, as we are solving a harder learning task.

6

Classical robust statistics and list-decodable estimation. Classical robust statistics dates back to Tukey and Huber [Tuk60, Hub64]. Computationally efficient robust estimators for high-dimensional Gaussians were first developed in [DKK+ 16, LRV16]. We use the sum-of-squares estimator of [KMZ22] and the bounded-covariance mean estimator of [DKP20]. List-decodable learning was introduced in [CSV17], and list-decodable estimation of Gaussian means and covariances was developed in [DKS18, IK22]. The classical literature on algorithmic robust statistic is too vast to do justice here, and we refer the reader to [DK23] for a textbook treatment. Gaussian versus non-Gaussian measurements. One notable feature of our results is a separation between what can be done for agnostic tomography of Gaussian states under Gaussian measurements versus nonGaussian measurements. Such separations for other tasks has a long history of study within quantum optics, for example the gap between Gaussian receivers and the Helstrom bound for discriminating coherent states [TS08], which non-Gaussian receivers based on photon counting and feedback can overcome [CMG07]. Other examples include bosonic data hiding against Gaussian operations and classical communication [SW21, Lam21], the dependence on number of modes for learning mixed Gaussian states [CMF+ 26], and the dependence on the energy for learning pure Gaussian states [CGYZ26].

2

Preliminaries

Throughout the paper, 𝐶 and 𝑐 denote sufficiently large and sufficiently small positive universal constants, respectively; their values may change between occurrences. Let 𝐷 denote the Frèchet derivative such that for function 𝑓 : 𝑉 → R, 𝐷 𝑘 𝑓 (𝜃) denotes the 𝑘-linear form where 𝐷 𝑘 𝑓 (𝜃) [ℎ1 , . . . , ℎ 𝑘 ] is the 𝑘-th derivative evaluated in the directions ℎ1 , . . . , ℎ 𝑘 ∈ 𝑉. For example, 𝐷 𝑓 (𝜃) [ℎ] ≡ 𝐷 ℎ 𝑓 (𝜃) is the directional derivative and 𝐷 2 𝑓 (𝜃) [ℎ1 , ℎ2 ] = ⟨ℎ1 , ∇2 𝑓 (𝜃)ℎ2 ⟩𝑉 is a quadratic form involving the Hessian.

2.1

Matrix decompositions

We say a matrix 𝐴 ∈ R2𝑛×2𝑛 is real symplectic if 𝐴Ω𝑛 𝐴⊤ = Ω𝑛 , where Ω𝑛 is the standard symplectic form: 

0 Ω𝑛 = −𝐼𝑛

 𝐼𝑛 . 0

Williamson decomposition A positive definite matrix 𝐴 ∈ R2𝑛×2𝑛 can be decomposed as 𝐴 = 𝑆(𝐼2 ⊗𝐷)𝑆 ⊤ , where 𝑆 is symplectic and 𝐷 is diagonal. The diagonal entries of 𝐷 are called the symplectic eigenvalues of 𝐴, and are always positive. Takagi factorization.

Every complex symmetric matrix 𝐴 = 𝐴⊤ admits a factorization

𝐴 = 𝑈𝑆𝑈 ⊤ ,

𝑈 †𝑈 = 𝐼,

𝑆 = diag(𝑠1 , . . . , 𝑠 𝑛 ),

The 𝑠 𝑗 are the singular values of 𝐴: indeed, 𝑈 ⊤𝑈¯ = 𝐼 gives 𝐴𝐴† = 𝑈𝑆 2𝑈 † .

7

𝑠 𝑗 ≥ 0.

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2.2

Bosonic quantum systems

Here, we briefly review some of the basics of bosonic quantum information. We will assume that the reader has some familiarity with this concept throughout this paper, and we refer the interested reader to [EP03, Ser23a] for more complete introduction to the topic. Fock space An 𝑛-mode bosonic system is a system on an infinite-dimensional Hilbert space. The Hilbert space of a single mode is H1 = 𝐿 2 (R), and the standard basis for this is the Fock basis F1 = {|𝑚⟩ : 𝑚 ∈ N}. The Hilbert space of 𝑛 modes H𝑛 is the tensor product of 𝑛 single-mode Hilbert spaces, with associated Fock basis F𝑛 = {|𝑚⟩ : 𝑚 ∈ N𝑛 }. Each mode has an associated annihilation and creation operator 𝑎ˆ 𝑖 , 𝑎ˆ †𝑖 , which can be defined by their action on the Fock basis: √︁ √ 𝑎ˆ †𝑖 |𝑚⟩ = 𝑚 𝑖 + 1 |𝑚 + 𝑒 𝑖 ⟩ , 𝑎ˆ 𝑖 |𝑚⟩ = 𝑚 𝑖 |𝑚 − 𝑒 𝑖 ⟩ , where 𝑒 𝑖 is the 𝑖-th standard basis vector. These operators satisfy the commutation relations [ 𝑎ˆ 𝑖 , 𝑎ˆ †𝑗 ] = 𝛿𝑖 𝑗 𝐼. These define the position and momentum quadrature field operators, respectively, given by R̂ = ( 𝑞ˆ 1 , . . . , 𝑞ˆ 𝑛 , 𝑝ˆ1 , . . . , 𝑝ˆ 𝑛 ) ⊤ 𝑎ˆ + 𝑎ˆ †

𝑎ˆ − 𝑎ˆ †

by 𝑞ˆ 𝑖 = 𝑖√ 𝑖 and 𝑝ˆ𝑖 = 𝑖 √ 𝑖 . Then, we have that [R̂𝑖 , R̂ 𝑗 ] = 𝑖Ω𝑛,𝑖 𝑗 . Also define the number operator 𝑖 2 Í𝑛 †2 𝑁 = 𝑖=1 𝑎ˆ 𝑖 𝑎ˆ 𝑖 . For a general 𝑛-mode bosonic state 𝜌, we can define its Wigner function 𝑊𝜌 : R2𝑛 → R by ∫ D 1 𝑦E 𝑛 𝑦 −𝑖 ( 𝑝·𝑦) 𝑊𝜌 (𝑞, 𝑝) = 𝑒 𝜌 𝑞 − 𝑑 𝑦. 𝑞 + (2𝜋) 𝑛 R𝑛 2 2 This is a quasi-probability distribution. Its mean is m and its covariance is 𝑉/2, where     m = m(𝜌) = tr 𝜌 R̂ , 𝑉 = 𝑉 (𝜌)𝑖 𝑗 = tr 𝜌{R̂𝑖 − m(𝜌)𝑖 , R̂ 𝑗 − m(𝜌) 𝑗 } . We refer to m as the mean or displacement vector of 𝜌, and 𝑉 is its covariance matrix; with this normalization, the vacuum has 𝑉 = 𝐼 and Wigner function N (0, 𝐼/2). 𝑉 is a real, symmetric matrix, and moreover always satisfies the following uncertainty relation 𝑉 + 𝑖Ω𝑛 ⪰ 0 ,

(3)

meaning that not all PSD covariance matrices are realizable. The energy operator and energy of a state can be written as 𝑛 1 1 ∑︁ 2 ( 𝑞ˆ 𝑗 + 𝑝ˆ2𝑗 ) = R̂⊤ R̂, 𝐸ˆ 𝑛 ≡ 2 𝑗=1 2

1 1 𝐸 (𝜌) ≡ tr 𝜌 𝐸ˆ 𝑛 = tr 𝑉 (𝜌) + ∥m(𝜌) ∥ 22 4 2

Í where we note that 𝐸ˆ 𝑛 = 𝑛𝑗=1 𝑎ˆ †𝑗 𝑎ˆ 𝑗 + 𝑛2 𝐼, meaning 𝐸 (𝜌) ≥ 𝑛/2, with equality for the vacuum. We also let 𝜈(𝜌) = tr(𝜌𝑁) = 𝐸 (𝜌) − 𝑛/2 denote the mean photon number of 𝜌.

8

Gaussian states A Gaussian state is the class of states whose Wigner function is a Gaussian density. Such states are uniquely determined by their mean and covariance. We will use m, 𝑉 to denote the mean and covariance of a quantum Gaussian state. A Gaussian state is pure if and only if 𝑉 = 𝑆𝑆 ⊤ for some symplectic matrix 𝑆 in the Williamson decomposition; that is, all of its symplectic eigenvalues are 1. One consequence is that the eigenvalues of the covariance matrix of a pure Gaussian state come in pairs (𝜆, 1/𝜆). We now record here a helpful bound for converting between various distances for bosonic Gaussian states and their parameters. Fact 2.1 (Theorem 8 [BMEM25]). Let 𝜋1 (m1 , 𝑉1 ) and 𝜋2 (m2 , 𝑉2 ) be two bosonic Gaussian states. Then,   𝑑tr (𝜋1 , 𝜋2 ) ≲ ∥𝑉1−1/2 (m1 − m2 ) ∥ 2 + tr (𝑉1−1 + 𝑉2−1 )|𝑉1 − 𝑉2 | . Gaussian unitary A Gaussian unitary is a unitary operator that maps every bosonic Gaussian state to Í ˆ another Gaussian state. Formally, a unitary 𝑈 is Gaussian if and only if 𝑈 † R̂ 𝑗 𝑈 = 2𝑛 𝑘=1 𝑆 𝑗 𝑘 Rk + 𝑑 𝑗 𝐼, 2𝑛×2𝑛 2𝑛 where 𝐼 denotes the identity operator, 𝑆 ∈ R is symplectic, and 𝑑 ∈ R is the displacement. Gaussian unitaries are generated by Hamiltonians which are quadratic in the quadrature operators, and thus physically realizable by standard techniques in quantum optics. Thus, throughout this paper, we will assume that given a symplectic matrix 𝑆 ∈ R2𝑛×2𝑛 and displacement 𝑑, one can efficiently apply its associated Gaussian unitary 𝑈 in polynomial time. For our purposes, the most important operational property of Gaussian unitaries is that if 𝑈 is a Gaussian unitary with associated symplectic matrix 𝑆 and displacement 𝑑, and 𝜌 is an arbitrary bosonic state with Wigner function 𝑊𝜌 , then the Wigner function of 𝜌 ′ = 𝑈 𝜌𝑈 † is given by 𝑊𝜌′ (𝑥) = 𝑊𝜌 (𝑆 −1 (𝑥 − 𝑑)) . In particular, this means a Gaussian state with mean m and covariance 𝑉 gets mapped to a new Gaussian state with mean 𝑆m + 𝑑 and covariance 𝑆𝑉 𝑆 ⊤ . Gaussian measurements Gaussian measurements are a powerful and implementable class of measurements on bosonic systems. These are POVMs of the form 𝑀 (𝑑 2𝑛 𝑦) =

1 𝐷 (𝑦)𝜏𝐷 (𝑦) † 𝑑 2𝑛 𝑦 , (2𝜋) 𝑛

where 𝜏 is a mean-zero Gaussian state known as the seed state, and its covariance 𝑊 is known as the seed covariance. Here 𝐷 (𝑦) is a displacement operator satisfying 𝐷 (𝑦) † R̂ 𝑗 𝐷 (𝑦) = R̂ 𝑗 + 𝑦 𝑗 𝐼 for all 𝑗 = 1, . . . , 2𝑛. Their key operational definition for our purposes is that when applied to a Gaussian state, the result is a draw from a classical Gaussian distribution. In particular, if we measure a Gaussian state with mean m and covariance 𝑉, the result is a draw from N (m, 𝑉+𝑊 2 ). When 𝑊 = 𝐼2𝑛 , this is called a heterodyne measurement. When 𝑊 = 𝐾 diag(𝑎, . . . , 𝑎, 𝑎 −1 , . . . , 𝑎 −1 )𝐾 ⊤ for an orthogonal symplectic matrix 𝐾 and 𝑎 > 0, it is a general-dyne measurement; in the limit 𝑎 → 0, its outcomes reduce to those of a homodyne measurement of the commuting quadratures 𝑘 1⊤ R̂, . . . , 𝑘 𝑛⊤ R̂, where 𝑘 𝑗 is the 𝑗-th column of 𝐾. Both of these are experimentally feasible, and so we will assume throughout this paper that we can implement such a measurement in polynomial time. More generally, by first applying a Gaussian unitary, and then applying

9

a heterodyne measurement, we obtain a general Gaussian measurement, which we assume can also be implemented in polynomial time. Energy and covariance. The operator norm of the covariance can be bounded in terms of the mean photon number as follows from [MBGF25]. Fact 2.2 (Lemma 37 of [MBGF25]). Let 𝜌 be an 𝑛-mode bosonic state with covariance matrix 𝑉 and finite mean photon number 𝜈 = 𝜈(𝜌). Then √︁ ∥𝑉 ∥ op ≤ 1 + 2𝜈 + 2 𝜈(𝜈 + 1) ≤ 4𝜈 + 2. In particular, for a pure Gaussian state 𝜋 with covariance 𝑉, ∥𝑉 −1 ∥ op = ∥𝑉 ∥ op ≤ 4𝜈(𝜋) + 2. Bargmann-Fock space. variables.

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The 𝑛-mode bosonic Fock space can be realized by holomorphic functions in 𝑛

Definition 2.3 (Bargmann-Fock Space). Define the following inner product for entire complex functions in 𝑛 variables: ∫ 2 −𝑛 ⟨ 𝑓 , 𝑔⟩ B𝑛 = 𝜋 𝑓¯(𝑧)𝑔(𝑧)𝑒 − ∥ 𝑧 ∥ 𝑑𝑧 C𝑛

Then, B𝑛 ≡ { 𝑓 : C𝑛 → C : ∥ 𝑓 ∥ B𝑛 < ∞} is a Hilbert space known as the Bargmann-Fock space. In particular, B𝑛 can be identified with the 𝑛-mode bosonic Fock space F𝑛 via the mappings: 𝑧 𝑘1 𝑧 𝑘2 · · · 𝑧 𝑛𝑘𝑛 |𝑘 1 𝑘 2 . . . 𝑘 𝑛 ⟩ ↔ √ 1 2 , 𝑘 1 !𝑘 2 ! · · · 𝑘 𝑛 !

𝑎ˆ †𝑖 ↔ 𝑧𝑖 ,

𝑎ˆ 𝑖 ↔ 𝜕𝑧𝑖

Importantly, this notation gives a unique parameterization for pure Gaussian states. Fact 2.4 (Bargmann Form). Every pure Gaussian state |𝜓⟩ can be uniquely written as   1 1 ⊤ ⊤ 𝑓 𝜓 (𝑧) = √︁ exp 𝑏 𝑧 + 𝑧 𝐴𝑧 2 𝑍 (𝑏, 𝐴) with 𝜃 = (𝑏, 𝐴) ∈ P𝑛 = {(𝑏, 𝐴) : 𝑏 ∈ C𝑛 , 𝐴 ∈ C𝑛×𝑛 , 𝐴 = 𝐴⊤ } and ∥ 𝐴∥ op < 1. The global phase is typically fixed so that the vacuum coefficient is positive. In Fock space, these coefficients parameterize the state as

10

follows: 𝑄𝐴

𝐿𝑏

}| { z }| { z  ∑︁  𝑛 𝑛 ∑︁ 1 1 |𝜓(𝑏, 𝐴)⟩ = √︁ exp 𝑏 𝑖 𝑎ˆ †𝑖 + 𝐴𝑖 𝑗 𝑎ˆ †𝑖 𝑎ˆ †𝑗 |0⟩ 2 𝑖, 𝑗=1 𝑍 (𝑏, 𝐴) 𝑖=1 | {z } = √︁



1

1+

𝑍 (𝑏, 𝐴)

𝑛 ∑︁ 𝑖=1

|Φ(𝑏, 𝐴) ⟩ 𝑛 ∑︁

1 𝑏 𝑖 𝑎ˆ †𝑖 + 2

( 𝐴𝑖 𝑗 + 𝑏 𝑖 𝑏 𝑗 ) 𝑎ˆ †𝑖 𝑎ˆ †𝑗 + · · ·

(5)

 |0⟩

(6)

𝑖, 𝑗=1

where 𝑍 (𝑏, 𝐴) is the normalization factor that gives |𝜓(𝑏, 𝐴)⟩ unit norm. We also define 𝐾 𝜃 ≡ 𝐿 𝑏 + 𝑄 𝐴 for 𝜃 = (𝑏, 𝐴). We first express the displacement and photon number directly in terms of the Bargmann parameters. In particular, the linear parameter 𝑏 is generally different from the physical displacement. Lemma 2.5 (Displacement and photon number in Bargmann coordinates). Let 𝐴 = 𝐴⊤ with ∥ 𝐴∥ op < 1, let |𝜓⟩ = |𝜓(𝑏, 𝐴)⟩ and 𝜋 𝑏, 𝐴 = |𝜓⟩⟨𝜓|, and set 𝛽 𝑗 = ⟨𝜓| 𝑎ˆ 𝑗 |𝜓⟩. If 𝑠1 , . . . , 𝑠 𝑛 are the singular values of 𝐴, then ¯ −1 (𝑏 + 𝐴 𝑏), ¯ 𝛽 = (𝐼 − 𝐴 𝐴)

¯ 𝑏 = 𝛽 − 𝐴 𝛽, (1 − ∥ 𝐴∥ op ) ∥ 𝛽∥ 2 ≤ ∥𝑏∥ 2 ≤ (1 + ∥ 𝐴∥ op ) ∥ 𝛽∥ 2 , 𝜈(𝜋 𝑏, 𝐴) = ∥ 𝛽∥ 22 +

𝑛 ∑︁

𝑠2𝑗

𝑗=1

1 − 𝑠2𝑗

(7) (8) (9)

.

We defer the proof to Appendix A.1.1. We next record a formula for the normalization 𝑍 (𝑏, 𝐴) = ∥Φ(𝑏, 𝐴) ∥ 2 in (5). The unnormalized ⊤ ⊤ vector to the Bargmann function 𝑓 (𝑧) = 𝑒 𝑏 𝑧+𝑧 𝐴𝑧/2 . Thus its squared norm is ∫ |Φ(𝑏, 𝐴)⟩− corresponds 2 𝜋 −𝑛 C𝑛 | 𝑓 (𝑧)| 2 𝑒 ∥ 𝑧 ∥ 2 𝑑 2𝑛 𝑧. Using |𝑒 𝑡 | 2 = 𝑒 2 Re 𝑡 gives 𝑍 (𝑏, 𝐴) = 𝜋 −𝑛

∫

2

⊤

⊤

𝑒 − ∥ 𝑧 ∥ 2 +2 Re(𝑏 𝑧)+Re(𝑧 𝐴𝑧) 𝑑 2𝑛 𝑧.

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C𝑛

The following lemma evaluates this integral, which lets us compute 𝑍 and its derivatives using real matrices. Lemma 2.6 (Gaussian integral for the normalization). For 𝐴 = 𝐴⊤ with ∥ 𝐴∥ op < 1, define 

 𝐼 − Re 𝐴 Im 𝐴 𝑀 ( 𝐴) = , Im 𝐴 𝐼 + Re 𝐴



 Re 𝑏 𝑢(𝑏) = . − Im 𝑏

If 𝑠1 , . . . , 𝑠 𝑛 are the singular values of 𝐴, then 𝑀 ( 𝐴) has eigenvalues 1 − 𝑠 𝑗 and 1 + 𝑠 𝑗 , and in particular is positive definite. Moreover, 1 log 𝑍 (𝑏, 𝐴) = − log det 𝑀 ( 𝐴) + 𝑢(𝑏) ⊤ 𝑀 ( 𝐴) −1 𝑢(𝑏). 2 We defer the proof to Appendix A.1.2. 11

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Parameter versus trace distance. with the inner product:

We will view P𝑛 as a real Hilbert space with dimension 𝑛2 +3𝑛 equipped 

1 ⟨(𝑏, 𝐴), (𝑏 , 𝐴 )⟩ P𝑛 = Re ⟨𝑏, 𝑏 ⟩ + tr( 𝐴∗ 𝐴′ ) 2 ′

′

′



such that ∥(𝑏, 𝐴) ∥ 2P = ∥𝑏∥ 22 + 12 ∥ 𝐴∥ 2𝐹 , where we drop the subscript when the mode count is implied. While this parameter distance is invariant to passive Gaussian unitaries, i.e. unitary changes of modes, this is not the case for general unitaries which mix creation and annihilation operators. For similar reasons, the parameter distance between two pure Gaussian states close in trace distance can be arbitrarily large. However, for states close to vacuum, parameter distance can be related to trace distance. Lemma 2.7. Let |𝜓(𝑏, 𝐴)⟩ denote a pure Gaussian state which is near-vacuum, in the sense that 𝑑tr (|𝜓⟩⟨𝜓| , |0⟩⟨0|) ≤ 𝜉0 for some universal constant 𝜉0 . Then, ∥ (𝑏, 𝐴) ∥ P ≲ 𝑑tr (|𝜓⟩⟨𝜓| , |0⟩⟨0|) . Proof. Define 𝜉 ≡ 𝑑tr (|𝜓⟩⟨𝜓| , |0⟩⟨0|) such that 𝜉 = √ 𝜉0 ≤ 1/ 2, expanding with (6) gives:

√︁

1 − |⟨0|𝜓(𝑏, 𝐴)⟩| 2 =

√︁ 1 − 1/𝑍 from (6). Taking

1 𝜉2 ∥ (𝑏, 𝐴 + 𝑏𝑏 ⊤ ) ∥ 2P = ∥𝑏∥ 22 + ∥ 𝐴 + 𝑏𝑏 ⊤ ∥ 2𝐹 ≤ 𝑍 − 1 = ≤ 2𝜉 2 2 1 − 𝜉2 where the left hand side is the |1⟩ , |2⟩ contribution to the norm in 𝑍 and where the right hand side removes the |0⟩ contribution. Then, √ √ ∥𝑏∥ 2 √ ∥ (𝑏, 𝐴) ∥ P ≤ ∥ (𝑏, 𝐴 + 𝑏𝑏 ⊤ ) ∥ P + ∥ (0, 𝑏𝑏 ⊤ ) ∥ P ≤ 2 𝜉 + √ 2 ≤ 2 𝜉 + 2 𝜉 2 ≲ 𝜉 2 which proves the claim.

□

We will use these near-vacuum properties in Section 3.4 for the iterative refinement step of the high-fidelity algorithm. Unsqueezing [BMEM25].

We state the deterministic Williamson estimate used in the iterative unsqueezing analysis of

Lemma 2.8 ([BMEM25, Lemma 12]). Let 𝑉 be the covariance matrix of an 𝑛-mode Gaussian state, and let 𝑉ˆ be a real symmetric matrix satisfying 0 ⪯ 𝑉ˆ − 𝑉 ⪯

2𝜁 (𝑉 + 𝐼) 1−𝜁

(12)

b ⊤ , where 𝑆 is symplectic and 𝐷 b = for some 𝜁 ∈ (0, 1). Take a Williamson decomposition 𝑉ˆ = 𝑆 𝐷𝑆 diag(𝑑1 , . . . , 𝑑 𝑛 , 𝑑1 , . . . , 𝑑 𝑛 ) with 𝑑 𝑗 > 0. Then the Gaussian unitary associated with 𝑆 −1 transforms 𝑉 into 𝑉 ′ = 𝑆 −1𝑉 𝑆 −⊤

12

satisfying √︄ ∥ (𝑉 ′ ) −1 ∥ op ≤

2.3

1+

2𝜁 ∥𝑉 −1 ∥ op + 1 . 1−𝜁

Primitives from robust statistics

We will also require two primitives from the literature on robust statistics; we again refer the reader to [DK23] for a more detailed exposition on this topic. Robustly learning a Gaussian Our first key primitive is the following guarantee for robustly learning a Gaussian, given samples from a distribution which is close in total variation distance to a Gaussian: Proposition 2.9 (Robustly Learning Gaussians, [DKK+ 16, DK23]). Let 𝑃 be a distribution over R𝑑 and 𝑄 = N (𝜇, Σ) be a Gaussian such that optTV ≡ 𝑑tv (𝑃, 𝑄) ≤ optTV for a sufficiently small universal constant optTV . Then, there exists an algorithm which given 𝑁 ≥ 𝑂 (𝑑 2 log(1/𝛿)/optTV 2 ) samples and poly(𝑁) time outputs estimates 𝜇ˆ and Σ̂, such that with probability at least 1 − 𝛿: ∥Σ −1/2 ( 𝜇ˆ − 𝜇) ∥ 2 ≤ 𝜁

and

∥Σ −1/2 Σ̂Σ −1/2 − 𝐼 ∥ 𝐹 ≤ 𝜁

(13)

for 𝜁 = 𝜁 (optTV ) = 𝑂 (optTV log 1/optTV ). This implies that 𝑑tv (N (𝜇, Σ), N ( 𝜇, ˆ Σ̂)) ≤ 𝑂 (optTV log 1/optTV ). List-decodable Gaussian estimation In the low fidelity regime, we will also need to learn non-trivial Gaussian approximations to classical distributions, even when the quality of the best Gaussian approximation is quite poor. In such cases, the best Gaussian approximation is no longer unique, and rather, the standard procedure is to output a small list of candidate parameters so that one of them is close to the true Gaussian. For us, we will require the results from [IK22] for list-decodable mean and covariance estimation, restated below. Lemma 2.10 ([IK22, Theorem 8.6]). There are universal constants 𝐶dec , 𝑟 dec > 1 such that the following holds. For 𝛽 ∈ (0, 1] and 𝑑 ≥ 1, set 𝑇𝛽 = 𝐶dec (2/𝛽) 𝑟dec ,

  𝑚 𝛽 (𝑑) = 𝑒𝑇𝛽 (𝑑 + 1) 𝑇𝛽 ,

ℓ𝛽 = ⌈𝑒𝑇𝛽 ⌉.

Given 𝛽 and 𝑚 𝛽 (𝑑) independent samples from any distribution 𝑃 on R𝑑 , a randomized algorithm with poly(𝑚 𝛽 (𝑑)) time returns a list of at most ℓ𝛽 pairs ( 𝜇, ˆ Σ̂) with Σ̂ ≻ 0. For every fixed Gaussian 𝑄 = N (𝜇, Σ) with Σ ≻ 0 and 𝑑tv (𝑃, 𝑄) ≤ 1 − 𝛽, with probability at least 9/10 the list contains a pair satisfying all three bounds 𝑇𝛽−1 Σ ⪯ Σ̂ ⪯ 𝑇𝛽 Σ, Σ −1/2 ( Σ̂ − Σ)Σ −1/2 𝐹 ≤ 𝑇𝛽2 , √︃ Σ −1/2 ( 𝜇ˆ − 𝜇) 2 ≤ 𝑇𝛽 (1 + 𝑇𝛽 ).

13

3

Agnostic learning in the high-fidelity regime

We begin in the high-fidelity regime, where the given state, 𝜌, is sufficiently close to a Gaussian state such that opt ≤ opt for some universal constant threshold opt. As an instructive exercise, we start by introducing the naive algorithm of heterodyne tomography and ask ourselves “how can we make this robust?”.

3.1

Robust heterodyne tomography

Heterodyne tomography, as the name suggests, applies a heterodyne measurement to 𝑁 copies of a Gaussian state 𝜋 with the hope of estimating parameters (m, 𝑉) using the resulting 𝑁 samples from N (m, (𝑉 + 𝐼2𝑛 )/2). As a first pass, this seems to follow immediately from re-arranging the empirical mean, 𝜇ˆ ℎ , and empirical covariance, Σ̂ℎ , to construct m̂ = 𝜇ˆ ℎ and 𝑉ˆ = 2Σ̂ℎ − 𝐼. However, this may fail to correspond with a valid Gaussian state since the covariance error 𝑉ˆ − 𝑉 is two-sided, causing a potential violation of the symplectic condition in (3). Thus, we modify our covariance estimator to first scale the empirical covariance according to its relative error guarantee such that 𝑉ˆ − 𝑉 is one-sided and the condition is satisfied. Proposition 3.1 (Heterodyne Tomography, Theorem 10 of [BMEM25]). Let {𝑦 𝑖 }𝑖 ∈ [ 𝑁 ] be the 𝑁 samples obtained from a heterodyne measurement on 𝜋, and let 𝜇ˆ ℎ , Σ̂ℎ be the empirical mean and covariance respectively such that the latter has relative error (1 − 𝜁)Σℎ ⪯ Σ̂ℎ ⪯ (1 + 𝜁)Σℎ . Construct the following estimators: 2Σ̂ℎ ˆ m̂ ≡ 𝜇ˆ ℎ , 𝑉ˆ ≡ − 𝐼, 𝜋ˆ ≡ 𝜋( m̂, 𝑉) (14) 1−𝜁 √︃ 𝑛+log 1/ 𝛿 For universal constant 𝐶, take 𝜁 = 𝐶 such that 𝜁 ≤ 1/4 for suitably large 𝑁. Then, with probability 𝑁 at least 1 − 𝛿, the relative error is at most √︂ 𝑛 + log 1/𝛿 −1 𝑑tr (𝜋, 𝜋) ˆ ≲ (2𝑛 + tr 𝑉 ) ≤ 𝜖 for 𝑁 = 𝑂 (𝑛(𝑛 + tr 𝑉 −1 ) 2 𝜖 −2 log 1/𝛿) (15) 𝑁 or alternatively 𝑁 = 𝑂 (𝑛𝐸 (𝜋) 2 𝜖 −2 log 1/𝛿). Unfortunately, heterodyne tomography is not robust. To illustrate this, suppose we receive copies of 𝜌 = (1 − 𝜂) |0⟩⟨0| + 𝜂 |𝜉⟩⟨𝜉 | where |𝜉⟩ = ( 𝑎ˆ †1 ) ℓ |0⟩ for some small 𝜂 ≤ opt and positive integer ℓ. Then, opt ≤ opt since 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝜂, but the outcome distribution of applying a heterodyne measurement to 𝜌 is (1 − 𝜂)N (0, 𝐼) + 𝜂N (ℓ𝑒 1 , 𝐼), whose mean 𝜂ℓ𝑒 1 explodes as ℓ → ∞. This failure comes from the well-known fact that the mean can be heavily skewed by outliers. Luckily, this issue is precisely the origin of robust statistics, a developed literature in classical statistics focused on building estimators which can withstand small corruptions. These tools give us a clear path toward a robust version of heterodyne tomography: perform a heterodyne measurement, use off-the-shelf methods to obtain robust mean and covariance estimates, and then substitute these into (14). We can then obtain the following robust tomography algorithm. Theorem 3.2 (Robust Heterodyne Tomography). Let 𝜌 be an 𝑛-mode state where 𝑑tr (𝜌, 𝜋) ≤ opttr ≤ opttr for some Gaussian state 𝜋 with mean m and covariance 𝑉. Then, there exists an algorithm which given 𝑂˜ (𝑛2 log5 (1/𝛿)/opttr 2 ) copies of 𝜌 constructs a Gaussian state 𝜋ˆ with the semi-agnostic guarantee: 𝑑tr (𝜌, 𝜋) ˆ ≲ (𝑛 + tr 𝑉 −1 ) · 𝑂˜ (opttr ) ≲ 𝐸 (𝜋) · 𝑂˜ (opttr ). 14

(16)

We defer the proof to Appendix A.2.1. Unfortunately, the semi-agnostic guarantee in (16) is quite disappointing since the error scales with the energy of the closest Gaussian state, which is at least linear in the number of modes, 𝑛. Thus, Theorem 3.2 is generally only useful when we are okay with mode factors inflating the original corruption size.

3.2

Robust iterative unsqueezing

While heterodyne tomography is simple, we incur a quadratic energy dependence in the sample complexity 𝑂 (𝑛𝐸 2 𝜖 −2 log 1/𝛿), which is very poor for squeezed states. Fortunately, subsequent work by [BMEM25] reduced the energy dependence to log log 𝐸 through a protocol which we call iterative unsqueezing. This involves a series of applications of heterodyne tomography to generate a sequence of Gaussian operations that unsqueeze the state, after which the resulting state is estimated with heterodyne tomography and then rotated back through the corresponding sequence of inverse Gaussian operations to derive the final estimate. Specifically, let 𝜋0 ≡ 𝜋 be the given Gaussian state, and conduct the following procedure for
𝐾 = log log ∥𝑉 −1 ∥ op total rounds: 1. Perform heterodyne tomography on 𝜋 𝑘 with roughly 𝑂 (𝑛) copies and failure probability 𝛿/𝐾 to arrive at estimates 𝜋ˆ 𝑘 = 𝜋( m̂ 𝑘 , 𝑉ˆ 𝑘 ). 2. Compute the Williamson decomposition of the covariance estimate 𝑉ˆ 𝑘 = 𝑆ˆ 𝑘 𝐷ˆ 𝑘 𝑆ˆ⊤𝑘 . 3. Form 𝜋 𝑘+1 = 𝑈𝑆ˆ −1 𝜌𝑈 †ˆ −1 which has the same mean as 𝜌 but covariance 𝑆 −1𝑉 𝑆 −1⊤ . 𝑘

𝑆𝑘

4. Repeat the previous steps for 𝑘 ← 𝑘 + 1 The intuition is that each step unsqueezes the incoming state until 𝜋 𝐾 has a covariance matrix whose inverse is bounded: ∥𝑉𝐾−1 ∥ op ≤ 2. Since no particular direction is exponentially small due to squeezing, this tames the tr 𝑉 −1 contribution in (15), meaning we can perform heterodyne tomography on 𝜋 𝐾 with 𝑂 (𝑛3 𝜖 −2 log 1/𝛿) copies: 4. Perform heterodyne tomography on 𝜋 𝐾 with roughly 𝑂 (𝑛3 /𝜖 2 ) copies to obtain 𝜋ˆ 𝐾 ≡ 𝜋( m̂𝐾 , 𝑉ˆ𝐾 ). 5. Form 𝜋ˆ = 𝑈 †ˆ −1 · · · 𝑈 †ˆ −1 𝜋ˆ 𝐾 𝑈𝑆ˆ −1 · · · 𝑈𝑆ˆ −1 as the final estimate. 𝑆1

𝑆𝐾

𝐾

1

The only energy dependence in the sample complexity comes from the number of iterations 𝐾 = log log ∥𝑉 −1 ∥ op ≤ log log 𝐸 (𝜋), with 𝑂 (𝑛) copies used for the heterodyne tomography of step 1 in each round. This gives the following net copy complexity. Theorem 3.3 (“Energy-Independent” Tomography via Iterative Unsqueezing, Theorem 3 [BMEM25]). Let 𝜋 be a Gaussian state with unknown covariance 𝑉. Then, there exists an algorithm which takes 𝑁 = 𝑂 (((𝑛 + log log log ∥𝑉 −1 ∥ op ) log log ∥𝑉 −1 ∥ op + 𝑛3 𝜖 −2 ) log 1/𝛿) copies of 𝜋 and uses exclusively Gaussian measurements to ouptut a Gaussian state 𝜋ˆ such that 𝑑tr (𝜋, 𝜋) ˆ ≤𝜖 with probability at least 1 − 𝛿.

15

Again, we ask if this “energy-independent” Gaussian state tomography algorithm can be robustified. The natural first suggestion is to run the algorithm on the given copies of 𝜌, constructing 𝜌ˆ 𝑘 , m̂ 𝑘 , 𝑉ˆ 𝑘 , 𝑆ˆ 𝑘 , 𝐷ˆ 𝑘 accordingly for 𝑘 ∈ [𝐾], by replacing every occurrence of heterodyne tomography in the algorithm with the robust primitive in Theorem 3.2. At first, this seems to immediately fail because the error of robust heterodyne tomography in step 1 for the first iteration blows up by a linear factor in 𝐸 (𝜋0 ), which may be arbitrarily large. Fortunately, this is not an issue because heterodyne tomography is only used for a covariance estimate with constant relative error, 𝜁 ≤ 1/4, during the iterative unsqueezing procedure. In fact, this is why each iteration uses only 𝑂 (𝑛) copies as opposed 𝑂 (𝑛𝐸 (𝜋) 2 𝜖 −2 log 𝐾/𝛿). Thus, by substituting the robust heterodyne tomography primitive, we can make this “energy-independent” protocol robust. Theorem 3.4 (Robust “Energy-Independent” Tomography via Iterative Unsqueezing). Let 𝜌 be a 𝑛-mode state where 𝑑tr (𝜌, 𝜋) ≤ opttr ≤ opttr for Gaussian state 𝜋 with covariance 𝑉. Then, there exists an algorithm which takes 𝑂 (𝑛3 opttr −2 log log ∥𝑉 −1 ∥ op log5 (1/𝛿)) copies of 𝜌 and uses exclusively Gaussian measurements to output a Gaussian state 𝜋ˆ satisfying 𝑑tr (𝜌, 𝜋) ˆ ≤ 𝑛 · 𝑂˜ (opttr ).

(17)

We defer the proof to Appendix A.2.2. While this robust iterative unsqueezing successfully removes the tr 𝑉 −1 factor in (16), the linear mode factor in (17) still gives only a semi-agnostic guarantee for arbitrary Gaussian states.

3.3

Robust pure state projection

Now, we do not expect to do better for mixed Gaussian states since the trace distance between such states is √ only bounded by parameter distances up to a necessary 𝑛 factor [CMF+ 26]. However, this mode factor disappears for pure Gaussian states, enabling [CMF+ 26] to optimally resolve tomography for this class; the algorithm is identical to Theorem 3.3, except the final unsqueezed state is projected to pure by taking the raw, uninflated covariance estimate and replacing all symplectic eigenvalues with 1. Specifically, if m̂𝐾 2Σ̂ℎ and 𝑉ˆ𝐾 = 1−𝜁 − 𝐼 are the estimates from step 5 of Theorem 3.3, we take the Williamson decomposition of ˆ where m̂ ≡ m̂𝐾 and 𝑉ˆ ≡ 𝑆ˆ 𝑆ˆ⊤ , before rotating back in step 6. 𝑉ˆraw ≡ 2Σ̂ℎ − 𝐼 = 𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ and form 𝜋( m̂, 𝑉) This gives the following guarantee. Theorem 3.5 (Near-Optimal Tomography for Pure Gaussian States, Theorem S7 [CMF+ 26]). Let |𝜓⟩ be a pure Gaussian state with covariance 𝑉. Then, there exists an algorithm which takes 𝑂 (𝑛2 /𝜖 2 + (𝑛 + log log log ∥𝑉 ∥ op ) log log ∥𝑉 ∥ op ) copies and uses exclusively Gaussian measurements to output a pure ˆ such that, with probability 2/3, 𝑑tr (|𝜓⟩⟨𝜓| , | 𝜓⟩⟨ ˆ 𝜓|) ˆ ≤ 𝜖. Gaussian state | 𝜓⟩ Again, we consider what this means for the robust setting: does appending pure state projection to Theorem 3.4 remove the 𝑛 factor in (17) to give a truly semi-agnostic guarantee? We respond in the affirmative with the following result. Theorem 3.6. Let 𝜌 be an 𝑛-mode state where 𝑑tr (𝜌, |𝜓⟩⟨𝜓|) ≤ opttr ≤ opttr for pure Gaussian state |𝜓⟩ with covariance 𝑉. Then, given 𝜖 > 0 and 𝛿 ∈ (0, 1/2), there exists an algorithm which takes 𝑂˜ (𝑛2 𝜖 −2 log log ∥𝑉 −1 ∥ op log5 (1/𝛿)) copies of 𝜌 and uses exclusively Gaussian measurements to output ˆ satisfying a pure Gaussian state | 𝜓⟩ ˆ ≤ 𝑂˜ (opttr ) + 𝜖 . ˆ 𝜓|) 𝑑tr (𝜌, | 𝜓⟩⟨ 16

(18)

with probability at least 1 − 𝛿. From Theorem 3.4, we have that ∥𝑉𝐾−1 ∥ op ≤ 2. Since trace distance is unitarily invariant, WLOG it suffices to just assume that ∥𝑉 −1 ∥ op ≤ 2 already and prove the guarantee using notation for the original state. We extract the bulk of this proof into the following lemma, whose proof closely follows that of Theorem S7 in [CMF+ 26], with additional care to remove the strong operator norm assumption in (S107) of [CMF+ 26] and instead control the relative Frobenius norm of the estimates. Lemma 3.7. Suppose ∥𝑉 −1 ∥ op ≤ 2 and that there exist estimators ( 𝜇ˆ ℎ , Σ̂ℎ ) which satisfy ∥ 𝜇ˆ ℎ − 𝜇 ℎ ∥ 2 ≤ 𝜁,

∥ Σ̂ℎ − Σℎ ∥ 𝐹 ≤ 𝜁

(19)

where 𝜇 ℎ = m and Σℎ = 𝐼+𝑉 2 denote the parameters of the heterodyne outcome distribution for pure Gaussian ˆ where 𝑉ˆ = 𝑆ˆ 𝑆ˆ⊤ from the decomposition of ˆ ≡ 𝜋( 𝜇ˆ ℎ , 𝑉) ˆ 𝜓| state |𝜓⟩⟨𝜓| = 𝜋(m, 𝑉). Define the estimate | 𝜓⟩⟨ 𝑉ˆraw ≡ 2Σ̂ℎ − 𝐼 = 𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ . Then, ˆ 𝜓| ˆ , |𝜓⟩⟨𝜓|) ≲ 𝜁 𝑑tr ( | 𝜓⟩⟨ provided that 𝜁 is sufficiently small. We defer the proof to Appendix A.2.3. Proof of Theorem 3.6. Run robust heterodyne tomography to obtain ( 𝜇ˆ ℎ , Σ̂ℎ ) which, by Proposition 2.9, satisfies ∥Σℎ−1/2 ( 𝜇ˆ ℎ − 𝜇 ℎ ) ∥ 2 ≤ 𝜁 and ∥Σℎ−1/2 Σ̂ℎ Σℎ−1/2 − 𝐼 ∥ 𝐹 ≤ 𝜁 (20) for 𝜁 = 𝑂˜ (opttr ) that satisfies 𝜁 ≤ 1/4. Since 12 𝐼 ⪯ 𝑉 ⪯ 2𝐼, 43 𝐼 ⪯ Σℎ ⪯ 32 𝐼, meaning (20) implies √︃ ∥ 𝜇ˆ ℎ − 𝜇 ℎ ∥ 2 ≤ 32 𝜁 and ∥ Σ̂ℎ − Σℎ ∥ 𝐹 ≤ 32 𝜁. Then, by triangle inequality, ˆ 𝜓|) ˆ ≤ 𝑑tr (𝜌, |𝜓⟩⟨𝜓|) + 𝑑tr (|𝜓⟩⟨𝜓| , | 𝜓⟩⟨ ˆ 𝜓|) ˆ ≤ opttr + 𝑂 (𝜁) = 𝑂˜ (opttr ) 𝑑tr (𝜌, | 𝜓⟩⟨ where we use the absolute guarantees to invoke Lemma 3.7 and bound the second term.

3.4

□

Robust iterative refinement via gradient descent

While the semi-agnostic guarantee in (18) does not contain mode or energy factors, it does lose logarithmic factors in opttr . This loss traces itself through the robust heterodyne tomography primitive back to the robust learner of Proposition 2.9 itself. As a result, any algorithm which simply applies robust heterodyne tomography should not expect to break the 𝑂˜ (opttr ) barrier. We circumvent this nevertheless by using non-Gaussian measurements to iteratively refine our estimate in Theorem 3.6 to reach not just 𝑂 (opt) but the optimal scaling of opt + 𝜖. We now state our main result. Theorem 3.8. Let 𝜌 be an arbitrary 𝑛-mode bosonic state and |𝜓⟩ be a pure Gaussian state with covariance 𝑉 such that 𝑑IF (𝜌, |𝜓⟩⟨𝜓|) ≤ opt ≤ opt for universal constant opt. Then, for given 𝜖 > 0 and 𝛿 ∈ (0, 1/2), there exists an algorithm which takes   𝑂˜ 𝑛2 (log log ∥𝑉 −1 ∥ op log5 (1/𝛿) + 𝜖 −1 )

17

ˆ satisfying copies and outputs a pure Gaussian state | 𝜓⟩ ˆ ≤ opt + 𝜖 ˆ 𝜓|) 𝑑IF (𝜌, | 𝜓⟩⟨ with probability at least 1 − 𝛿. Our algorithm begins by running the robust learner from Theorem 3.6 to obtain the pure Gaussian state | 𝜓ˆ 0 ⟩. However, since Theorem 3.6 requires closeness in trace distance despite Theorem 3.8 assuming closeness in infidelity, we must apply the Fuchs-van de Graaf inequality. Specifically, 𝑑tr (𝜌, |𝜓⟩⟨𝜓|) ≤ √︁ √ √ 1 − 𝐹 (𝜌, |𝜓⟩⟨𝜓|) ≤ opt, so the guarantee of Theorem 3.6 applies with opttr = opt, meaning 𝑑tr (𝜌, | 𝜓ˆ 0 ⟩⟨𝜓ˆ 0 |) ≤ √ 𝛾 where 𝛾 = 𝑂˜ ( opt) + 𝜖 0 . Re-writing in infidelity, √︃ 1 − ⟨𝜓ˆ 0 | 𝜌 | 𝜓ˆ 0 ⟩ = ∥ (𝐼 − 𝜌) 1/2 | 𝜓ˆ 0 ⟩ ∥ 2 ≤ ∥ (𝐼 − 𝜌) 1/2 |𝜓⟩ ∥ 2 + ∥ (𝐼 − 𝜌) 1/2 ( | 𝜓ˆ 0 ⟩ − |𝜓⟩) ∥ 2 √︁ ≤ 1 − ⟨𝜓| 𝜌 |𝜓⟩ + ∥ | 𝜓ˆ 0 ⟩ − |𝜓⟩ ∥ 2 √ √︁ ≤ opt + 2 𝑑tr (|𝜓⟩⟨𝜓| , | 𝜓ˆ 0 ⟩⟨𝜓ˆ 0 |) √ √︁ √︁ ≤ opt + 2( opt + 𝛾) 𝑑IF (𝜌, | 𝜓ˆ 0 ⟩⟨𝜓ˆ 0 |) ≤ 𝑂˜ (opt) + 𝑂 (𝜖 2 )

(21)

0

√ where we use triangle inequality and the facts that 𝐼 − 𝜌 ⪯ 𝐼 and ∥ | 𝜓ˆ 0 ⟩ − |𝜓⟩ ∥ 2 ≤ 2𝑑tr (|𝜓⟩⟨𝜓| , | 𝜓ˆ 0 ⟩⟨𝜓ˆ 0 |). If 𝑈0 is the Gaussian unitary satisfying 𝑈0† | 𝜓ˆ 0 ⟩ = |0⟩, by the unitary invariance of trace distance and infidelity, we can apply 𝑈0† to any 𝜌 we use in this stage, run our refinement algorithm, and then perform 𝑈0 𝜋𝑈 ˆ 0† to rotate back. WLOG, it then suffices to ignore the first learner and simply assume that the given state is near-vacuum: 𝑑IF (𝜌, |0⟩⟨0|) ≤ 𝑂˜ (opt) + 𝑂 (𝜖 02 ),

√︁ 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝛾 = 𝑂˜ ( opt) + 𝜖 0

This allows us to leverage properties of near-vacuum Gaussian states, such as Lemma 2.7, to study the geometry of the infidelity objective 𝑓𝜌 (𝜃) = 1 − ⟨𝜓(𝜃)| 𝜌 |𝜓(𝜃)⟩ as a function of the Bargmann parameters expressed as reals: 𝜃 = (𝑏, 𝐴) ∈ P. For |𝜓(𝜃)⟩ near vacuum, i.e. 𝜃 ∈ 𝐵𝑟 = {𝜃 ∈ P : ∥𝜃 ∥ P ≤ 𝑟 } for sufficiently small 𝑟, we find that 𝑓𝜌 (𝜃) is smooth and strongly convex. Thus, our algorithm proceeds by performing gradient descent on 𝑓𝜌 (𝜃); we will use measurements to simulate noisy access to the objective 𝑓𝜌 (𝜃), which we then use to simulate access to the gradient via finite differencing. 3.4.1

Near-vacuum geometry of the infidelity

We find that the Hessian of 𝑓𝜌 (𝜃) can be controlled by the trace distance to the vacuum state, provided this quantity sufficiently small. Proposition 3.9. There exist universal constants 𝑟 0 , 𝐶 > 0 such that for every 𝑛, every density operator 𝜌, and every 𝜃 ∈ 𝐵𝑟0 ,
∥∇2 𝑓𝜌 (𝜃) − 2𝐼 ∥ op ≤ 𝐶 ∥𝜃 ∥ P + 𝑑tr (𝜌, |0⟩⟨0|) . (22) 18

where the operator norm is induced by ∥ · ∥ P . In particular, there are universal constants 𝛾0 > 0 and 𝑅 ∈ (0, 1/16) such that, whenever 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝛾0 , the infidelity objective satisfies 𝐼 ⪯ ∇2 𝑓𝜌 (𝜃) ⪯ 3𝐼

(𝜃 ∈ 𝐵 𝑅 ),

(23)

and its unique minimizer 𝜃 ∗ over 𝐵 𝑅 satisfies ∥𝜃 ∗ ∥ P ≤ 𝑅/4. We make headway toward (22) by first controlling higher-order derivative tensors of the state |𝜓(𝜃)⟩ nearvacuum. We start by understanding how the operator in the exponent of (5), 𝐾 𝜃 , inflates the norm of various states in Fock space. Lemma 3.10. Let Π ≤𝑚 be the projector onto states with total photon number at most 𝑚. For |𝜙⟩ ∈ F≤𝑚 ≡ Π ≤𝑚 F , we have that ∥𝐾 𝜃 |𝜙⟩ ∥ 2 ≤ 2(𝑚 + 2) ∥𝜃 ∥ P ∥ |𝜙⟩ ∥ 2 . Proof. WLOG, we perform a unitary change of modes such that 𝑏 = ∥𝑏∥ 2 𝑒 1 . Then, ∥𝐿 𝑏 𝜙∥ 22 = ∥𝑏∥ 22 ⟨𝜙| 𝑎ˆ 1 𝑎ˆ †1 |𝜙⟩ = ∥𝑏∥ 22 ⟨𝜙| ( 𝑎ˆ †1 𝑎ˆ 1 + 𝐼) |𝜙⟩ ≤ (𝑚 + 1) ∥𝑏∥ 22 ∥ |𝜙⟩ ∥ 22 Similarly, we claim that: ∥𝑄 𝐴 |𝜙⟩ ∥ 2 ≤

√︁ 1 ∥ 𝐴∥ 𝐹 (𝑚 + 1) (𝑚 + 2) ∥ |𝜙⟩ ∥ 2 2

(24)

Since every term in 𝑄 𝐴 has two creation operators and |𝜙⟩ ∈ F≤𝑚 , we have that 𝑄 𝐴 |𝜙⟩ ∈ F≤𝑚+2 . Considering the variational characterization of the left hand side for any normalized |𝜑⟩ ∈ F≤𝑚+2 , 1/2 𝑛 𝑛 1 ∑︁ 1 © ∑︁ ª † † | ⟨𝜑| 𝑄 𝐴 |𝜙⟩ | = 𝐴𝑖 𝑗 ⟨𝜑| 𝑎ˆ 𝑖 𝑎ˆ 𝑗 |𝜙⟩ ≤ ∥ 𝐴∥ 𝐹 ­ | ⟨𝜑| 𝑎ˆ †𝑖 𝑎ˆ †𝑗 |𝜙⟩ | 2 ® 2 𝑖, 𝑗=1 2 «𝑖, 𝑗=1 ¬

1/2

𝑛 1 © ∑︁ ª ≤ ∥ 𝐴∥ 𝐹 ∥ |𝜙⟩ ∥ 2 ­ ∥ 𝑎ˆ 𝑗 𝑎ˆ 𝑖 |𝜑⟩ ∥ 22 ® 2 «𝑖, 𝑗=1 ¬

(25)

where we apply Cauchy-Schwarz twice. Since the ladder operators for different modes commute, for 𝑖 ≠ 𝑗, 𝑎ˆ †𝑖 𝑎ˆ †𝑗 𝑎ˆ 𝑗 𝑎ˆ 𝑖 = 𝑁𝑖 𝑁 𝑗 where 𝑁𝑖 = 𝑎ˆ †𝑖 𝑎ˆ 𝑖 is the number operator for the 𝑖-th mode, while for 𝑖 = 𝑗, 𝑎ˆ †𝑖 𝑎ˆ †𝑗 𝑎ˆ 𝑗 𝑎ˆ 𝑖 = 𝑎ˆ †𝑖 ( 𝑎ˆ 𝑖 𝑎ˆ †𝑖 − 1) 𝑎ˆ 𝑖 = 𝑁𝑖 (𝑁𝑖 − 1). Then, 𝑛 ∑︁

𝑛 ª © ∑︁ † † 2 𝑎 𝑖 𝑎ˆ 𝑗 𝑎ˆ 𝑗 𝑎ˆ 𝑖 ® |𝜑⟩ = ⟨𝜑| ∥ 𝑎ˆ 𝑗 𝑎ˆ 𝑖 |𝜑⟩ ∥ 2 = ⟨𝜑| ­ 𝑖, 𝑗=1 ¬ «𝑖, 𝑗=1

𝑛 ∑︁ 𝑖≠ 𝑗

𝑁𝑖 𝑁 𝑗 +

𝑛 ∑︁

! 𝑁𝑖 (𝑁𝑖 − 1) |𝜑⟩ = ⟨𝜑| 𝑁 (𝑁 − 1) |𝜑⟩

𝑖=1

≤ (𝑚 + 2) (𝑚 + 1)

Í𝑛 where 𝑁 = 𝑖=1 𝑁𝑖 is the total number operator and where we recall that |𝜑⟩ ∈ F≤𝑚+2 in the last equality. Substituting this into (25) then proves (24). Thus, by triangle inequality,   √︁ √ 1 ∥𝐾 𝜃 |𝜙⟩ ∥ 2 ≤ ∥𝐿 𝑏 |𝜙⟩ ∥ 2 + ∥𝑄 𝐴 |𝜙⟩ ∥ 2 ≤ ∥𝑏∥ 2 𝑚 + 1 + ∥ 𝐴∥ 𝐹 (𝑚 + 1) (𝑚 + 2) ∥ |𝜙⟩ ∥ 2 2 ≤ 2(𝑚 + 2) ∥𝜃 ∥ P ∥ |𝜙⟩ ∥ 2 19

which proves the bound.

□

Control on the norm growth that 𝐾 𝜃 induces allows us to bound the magnitudes of higher-order derivatives of |𝜓(𝜃)⟩ by carefully differentiating and tracking the total photon number of each intermediate state. Lemma 3.11. There is a universal constant 𝐶𝐷 such that, for every 𝑘 ≥ 1, ∥𝜃 ∥ P ≤ 1/8, and directions ℎ1 , . . . , ℎ 𝑘 ∈ P, 𝑘 Ö 𝑘 𝑘 ∥ℎ𝑖 ∥ P . ∥ 𝐷 𝜓(𝜃) [ℎ1 , . . . , ℎ 𝑘 ] ∥ 2 ≤ 𝐶𝐷 𝑘! 𝑖=1

We defer the proof to Appendix A.2.4. This higher-order derivative control on |𝜓(𝜃)⟩ is sufficient to prove Proposition 3.9. Proof of Proposition 3.9. Let 𝑟 0 = 1/8, and for brevity, denote 𝑔 ℎ (𝜃) ≡ 𝐷𝑔(𝜃) [ℎ] and 𝑔 ℎ1 ℎ2 (𝜃) ≡ 𝑗 𝐷 2 𝑔(𝜃) [ℎ1 , ℎ2 ], where 𝑔 may be 𝑍, Φ, or 𝜓. Further, let 𝐶 𝑗 ≡ 𝐶𝐷 𝑗!. In broad strokes, we first compute the Hessian of 𝑓 |0⟩⟨0| (𝜃) at 𝜃 = 0. Then, we use third derivative control to control the Hessian of 𝑓 |0⟩⟨0| (𝜃) for ∥𝜃 ∥ P ≤ 𝑟 0 , and finally, we use second derivative control to handle the Hessian of 𝑓𝜌 (𝜃). We begin with the Hessian for 𝜌 = |0⟩⟨0| at 𝜃 = 0. Since 𝐾 𝜃 is linear in 𝜃 and all creation operators commute, 𝑒 𝐾 𝜃+𝑡 ℎ = 𝑒 𝐾 𝜃 𝑒 𝑡 𝐾ℎ , so 𝐷Φ(𝜃) [ℎ] = 𝐾 ℎ 𝑒 𝐾 𝜃 |0⟩. With (5), this gives |Φ(0)⟩ = |0⟩ and 𝐷Φ(0) [ℎ] = 𝐾 ℎ |0⟩, which is orthogonal to |0⟩. Then, by product rule, 𝑍 ℎ (0) = 2 Re ⟨Φ(0)|Φℎ (0)⟩ = 0 ( 1 (((( −3/2 −1/2 ((𝑍( |𝜓 ℎ (0)⟩ = − (𝑍( |Φℎ (0)⟩ = 𝐾 ℎ |0⟩ (0) ( ℎ (0) |Φ(0)⟩ + 𝑍 (0) (2 Applying product rule again, 𝑓 |0⟩⟨0| (0) = 0  ⟨0|𝜓 𝐷 𝑓 |0⟩⟨0| (0) [ℎ] = −2 Re ⟨𝜓(0)|0⟩   ℎ (0)⟩ = 0 

(26)

2 2 2 (((2 (( ⟨0|𝜓 2∥( 𝐷 𝑓 |0⟩⟨0| (0) [ℎ, ℎ] = −2 Re ⟨𝜓(0)|0⟩ ⟨0|𝜓 ℎℎ (0)⟩ − ( ℎ (0)⟩ ∥ 2 = 2∥ |𝜓 ℎ (0)⟩ ∥ 2 = 2∥𝐾 ℎ |0⟩ ∥ 2 = 2∥ℎ∥ P 2

(27) where the second equality in the last line uses the following manipulation of the normalization ∥ |𝜓(𝜃)⟩ ∥ 2 = 1: 0 = 𝐷 2 ∥ |𝜓(𝜃)⟩ ∥ 22 [ℎ, ℎ] = 2 Re ⟨𝜓(𝜃)|𝜓 ℎℎ (𝜃)⟩ + 2∥ |𝜓 ℎ (𝜃)⟩ ∥ 22 With the vacuum Hessian at 𝜃 = 0 computed, we continue to control the change in the Hessian for ∥𝜃 ∥ P ≤ 𝑟 0 and then for 𝜌 beyond the vacuum state. Writing 𝑠 (𝐻 ) (𝜃) = ⟨𝜓(𝜃)| 𝐻 |𝜓(𝜃)⟩ for any bounded self-adjoint 𝐻, product rule yields: (𝐻 ) 𝑠 ℎℎ (𝜃) = 2 Re ⟨𝜓 ℎℎ (𝜃)| 𝐻 |𝜓(𝜃)⟩ + 2 ⟨𝜓 ℎ (𝜃)| 𝐻 |𝜓 ℎ (𝜃)⟩ (𝐻 ) 𝑠𝑢ℎℎ (𝜃) = 2 Re ⟨𝜓𝑢ℎℎ (𝜃)| 𝐻 |𝜓(𝜃)⟩ + 2 Re ⟨𝜓 ℎℎ (𝜃)| 𝐻 |𝜓𝑢 (𝜃)⟩ + 4 Re ⟨𝜓𝑢ℎ (𝜃)| 𝐻 |𝜓 ℎ (𝜃)⟩

20

Since ∥ |𝜓(𝜃)⟩ ∥ 2 = 1 and Lemma 3.11 holds for ∥𝜃 ∥ P ≤ 1/8 = 𝑟 0 , by triangle inequality and these bounds, (𝐻 ) |𝑠 ℎℎ (𝜃)| ≤ 2(𝐶2 + 𝐶12 ) ∥𝐻 ∥ op ∥ℎ∥ 2P (𝐻 ) |𝑠𝑢ℎℎ (𝜃)| ≤ (2𝐶3 + 6𝐶1𝐶2 ) ∥𝐻 ∥ op ∥𝑢∥ P ∥ℎ∥ 2P

(28)

Then, by the fundamental theorem of calculus, (27), and (28), |𝐷

2

𝑓 |0⟩⟨0| (𝜃) [ℎ, ℎ] − 2∥ℎ∥ 2P | =

∫ 1 0

𝐷 3 𝑓 |0⟩⟨0| (𝑡𝜃) [𝜃, ℎ, ℎ]𝑑𝑡 ≤ (2𝐶3 + 6𝐶1𝐶2 ) ∥𝜃 ∥ P ∥ℎ∥ 2P

(29)

where we notice 𝑓 |0⟩⟨0| (𝜃) = 1 − 𝑠 (𝐻 ) (𝜃) for 𝐻 = |0⟩⟨0| which has ∥𝐻 ∥ op = 1. Now all that is left is controlling the Hessian when Δ ≡ 𝜌 − |0⟩⟨0| ≠ 0. (Δ) |𝐷 2 𝑓𝜌 (𝜃) [ℎ, ℎ] − 𝐷 2 𝑓 |0⟩⟨0| (𝜃) [ℎ, ℎ] | = |𝑠 ℎℎ (𝜃)| ≤ 2(𝐶2 + 𝐶12 ) ∥Δ∥ op ∥ℎ∥ 2P

(30)

Noting that ∥Δ∥ op ≤ 2𝑑tr (𝜌, |0⟩⟨0|), by triangle inequality, (29), (30), |𝐷 2 𝑓𝜌 (𝜃) [ℎ, ℎ] − 2∥ℎ∥ 2P | ≤ (2𝐶3 + 6𝐶1𝐶2 ) ∥𝜃 ∥ P ∥ℎ∥ 2P + 2(𝐶2 + 𝐶12 ) ∥Δ∥ op ∥ℎ∥ 2P ≤ 𝐶 (∥𝜃 ∥ P + 𝑑tr (𝜌, |0⟩⟨0|)) ∥ℎ∥ 2P where we choose 𝐶 according to 𝐶1 , 𝐶2 , 𝐶3 . Since this holds for all unit ℎ, we have proven (22). Then, taking 𝑅 = min{𝑟 0 /4, 1/(2𝐶)}, 𝛾0 = min{1/(2𝐶), 𝑅 2 /64}, √ √ we have ∥∇2 𝑓𝜌 (𝜃) − 2𝐼 ∥ op ≤ 𝐶 (𝑅 + 𝛾0 ) ≤ 1 for 𝜃 ∈ 𝐵 𝑅 , proving (23). Further, ∥ 𝐴∥ op ≤ 2∥𝜃 ∥ P ≤ 2𝑅 < 1 for 𝑅 < 1/16, so every point of 𝐵 𝑅 describes a valid pure Gaussian state. To show that the last part about the minimizer 𝜃 ∗ over 𝐵 𝑅 , we use the fundamental theorem of calculus twice and recall that 𝑓 |0⟩⟨0| (0) = 𝐷 𝑓 |0⟩⟨0| (0) [𝜃] = 0 from (26): ∫ 1∫ 𝑠 𝑓 |0⟩⟨0| (𝜃) =

∫ 1

2

𝐷 𝑓 |0⟩⟨0| (𝑡𝜃) [𝜃, 𝜃]𝑑𝑡𝑑𝑠 = 0

0

2

∫ 1

(1 − 𝑡)𝐷 𝑓 |0⟩⟨0| (𝑡𝜃) [𝜃, 𝜃]𝑑𝑡 ≥ 0

0

(1 − 𝑡) ∥𝜃 ∥ 2P 𝑑𝑡 ≥

1 ∥𝜃 ∥ 2P 2

where we use the lower bound of (23). Then, for 𝜃 on the boundary of 𝐵 𝑅 , by triangle inequality, 𝛾0 ≤ 𝑅 2 /64, and ∥Δ∥ op ≤ 2𝛾0 , 𝑓𝜌 (𝜃) ≥ 𝑓 |0⟩⟨0| (𝜃) − | 𝑓𝜌 (𝜃) − 𝑓 |0⟩⟨0| (𝜃)| ≥ 𝑅2 /2 − ∥Δ∥ op ≥ 𝑅 2 /2 − 2𝛾0 ≥ 3𝑅 2 /8>2𝛾0 ≥ 𝑓𝜌 (0) where we use the fact that 𝑓𝜌 (0) = | ⟨0| (|0⟩⟨0| − 𝜌) |0⟩ | ≤ ∥Δ∥ op ≤ 2𝛾0 . Then, by continuity of 𝑓𝜌 , compactness of 𝐵 𝑅 , and the fact that the boundary is strictly worse than the origin, it follows that all minimizers are in the interior. Letting one minimizer be 𝜃 ∗ such that ∇ 𝑓𝜌 (𝜃 ∗ ) = 0, by the fundamental theorem of calculus and (23) in the same manner as before, (((( 𝑓𝜌 (𝜃) ≥ 𝑓𝜌 (𝜃 ∗ ) + ( 𝐷( 𝑓𝜌 ( (𝜃( ∗ ) [𝜃 − 𝜃 ∗ ] +

∫ 1 0

1 (1 − 𝑡)𝐷 2 𝑓𝜌 (𝜃 ∗ (1 − 𝑡) + 𝑡𝜃) [𝜃 − 𝜃 ∗ , 𝜃 − 𝜃 ∗ ]𝑑𝑡 ≥ 𝑓𝜌 (𝜃 ∗ ) + ∥𝜃 − 𝜃 ∗ ∥ 2P 2

21

which implies that any 𝜃 ≠ 𝜃 ∗ has 𝑓𝜌 (𝜃) > 𝑓𝜌 (𝜃 ∗ ), meaning the minimizer is unique. Finally, taking 𝜃 = 0, 1 2 2 □ 2 ∥𝜃 ∗ ∥ P ≤ 𝑓𝜌 (0) ≤ 2𝛾0 ≤ 𝑅 /32 proves the ∥𝜃 ∗ ∥ P ≤ 𝑅/4 claim. 3.4.2

Noisy objective estimate

Given that the objective function 𝑓𝜌 (𝜃) is both smooth and strongly convex from Proposition 3.9, all we need for gradient descent is to give a protocol for estimating both 𝑓𝜌 (𝜃) and ∇ 𝑓𝜌 (𝜃) given 𝜃 and copies of 𝜌. We do this with a single Gaussian unitary followed by an on-off photon detector as follows. Lemma 3.12. For 𝜃 ∈ 𝐵 𝑅 , we can estimate 𝑓𝜌 (𝜃) to additive error 𝜅 with probability at least 1 − 𝛿 using 𝑂 (𝜅 −2 log(2/𝛿)) copies of 𝜌. Proof. Let 𝑈 𝜃 be the Gaussian unitary which prepares |𝜓(𝜃)⟩ = 𝑈 𝜃 |0⟩. For each copy, we apply 𝑈 †𝜃 and then measure the two-outcome POVM {|0⟩⟨0| , 𝐼 − |0⟩⟨0|}. The latter outcome has the probability:   𝑝 𝐼 − |0⟩⟨0| = tr (𝐼 − |0⟩⟨0|)𝑈 †𝜃 𝜌𝑈 𝜃 = 𝑑IF (𝜌, |𝜓(𝜃)⟩⟨𝜓(𝜃)|) which is exactly the estimand. Thus, an empirical average with 𝑂 (𝜅 −2 log(2/𝛿)) copies is sufficient by Hoeffding’s inequality. □ We use this estimation protocol to construct a zeroth-order estimate for ∇ 𝑓𝜌 (𝜃) via finite differencing. Lemma 3.13. If 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝛾0 , 𝜃 ∈ 𝐵 𝑅/2 , and 𝜅 ∈ (0, 𝑅/4], we can estimate ∇ 𝑓𝜌 (𝜃) to additive error 𝜅 in ∥ · ∥ P with probability at least 1 − 𝛿 using 𝑂 (𝑛6 𝜅 −4 log(𝑛/𝛿)). We defer the proof to Appendix A.2.5. 3.4.3

Gradient descent

We now present and analyze the gradient descent algorithm for near-vacuum states. Proposition 3.14. Suppose 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝛾0 and let opt ≡ inf | 𝜓⟩ ∈ G 𝑑IF (𝜌, |𝜓⟩⟨𝜓|). Then, given 𝜖 ∈ (0, 1) and 𝛿 ∈ (0, 1/2), there exists an algorithm which takes 𝑂˜ (𝑛6 𝜖 −2 log(1/𝛿)) copies of 𝜌 and outputs a pure ˆ such that Gaussian state | 𝜓⟩ ˆ 𝜓|) ˆ ≤ opt + 𝜖 𝑑IF (𝜌, | 𝜓⟩⟨ with probability at least 1 − 𝛿. Proof. We first show that opt = min 𝜃 ∈ 𝐵𝑅 𝑑IF (𝜌, |𝜓(𝜃)⟩⟨𝜓(𝜃)|), i.e. the minimizer is near-vacuum. Let |𝜓⟩ be any pure Gaussian state with 𝑑IF (𝜌, |𝜓⟩⟨𝜓|) ≤ 𝑑IF (𝜌, |0⟩⟨0|). Then, since 𝑑IF (𝜌, |0⟩⟨0|) ≤ 2𝑑tr (𝜌, |0⟩⟨0|), √︁ √︁ 𝜉 ≡ 𝑑tr (|𝜓⟩⟨𝜓| , |0⟩⟨0|) ≤ 𝑑tr (𝜌, |𝜓⟩⟨𝜓|) + 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝑑IF (𝜌, |0⟩⟨0|) + 𝛾0 ≤ 2𝛾0 + 𝛾0 ≤ 𝑅/4 √ Then, provided 𝑅 ≤ 4𝜉0 , the proof of Lemma 2.7 gives that ∥𝜃 ∥ P ≤ 2(𝜉 + 𝜉 2 ) ≤ 2𝜉 ≤ 𝑅/2. This means that every pure Gaussian state with infidelity at least as good as the vacuum, including the minimizer, must lie in 𝐵 𝑅 . Thus, opt = min 𝑓𝜌 (𝜃) = 𝑓𝜌 (𝜃 ∗ ), (31) 𝜃 ∈ 𝐵𝑅

22

√︁ We begin our algorithm by taking 𝜅 = min{𝑅/4, 𝜖/6} and 𝑇 = max{1, log3/2 (𝑅/(4𝜅))}. Starting with 𝜃 0 = 0, we then iterate the following gradient descent step for 𝑡 ∈ [𝑇]: 1d 𝜃 𝑡 = 𝜃 𝑡 −1 − ∇ 𝑓𝜌 (𝜃 𝑡 ) 3

(32)

d where the step size is 1/3 and ∇ 𝑓𝜌 (𝜃 𝑡 ) is the zeroth-order estimator from Lemma 3.13. After the last step, ˆ = |𝜓(𝜃 𝑇 )⟩. we simply output | 𝜓⟩ We first use our Hessian bounds to show that the iterates approach 𝜃 ∗ . Since ∇ 𝑓𝜌 (𝜃 ∗ ) = 0, the fundamental theorem of calculus gives: ∫ 1 ∇ 𝑓𝜌 (𝜃 𝑡 ) = 𝐻 𝜃𝑡 (𝜃 𝑡 − 𝜃 ∗ ),

𝐻 𝜃𝑡 =

∇2 𝑓𝜌 ((1 − 𝑠)𝜃 ∗ + 𝑠𝜃 𝑡 )𝑑𝑠

(33)

0

where (23) implies 𝐼 ⪯ 𝐻 𝜃𝑡 ⪯ 3𝐼 and thus ∥𝐼 − 𝐻 𝜃𝑡 /3∥ op ≤ 2/3. Then, by this fact, triangle inequality, (32), (33), and the additive error in (3.13), 1 𝜅 1 d ∥𝜃 𝑡 − 𝜃 ∗ ∥ P ≤ 𝜃 𝑡 −1 − 𝜃 ∗ − ∇ 𝑓𝜌 (𝜃 𝑡 −1 ) + ∥∇ 𝑓𝜌 (𝜃 𝑡 −1 ) − ∇ 𝑓𝜌 (𝜃 𝑡 −1 ) ∥ ≤ ∥ (𝐼 − 𝐻 𝜃𝑡 −1 ) (𝜃 𝑡 −1 − 𝜃 ∗ ) ∥ P + 3 3 3 P ≤

2 𝜅 ∥𝜃 𝑡 −1 − 𝜃 ∗ ∥ P + 3 3

(34)

Since ∥𝜃 0 − 𝜃 ∗ ∥ P ≤ 𝑅/4 by Proposition 3.9, if ∥𝜃 𝑡 −1 − 𝜃 ∗ ∥ P ≤ 𝑅/4, 𝜃 𝑡 ∈ 𝐵 𝑅/2 . Substituting into (34) with 𝜅 ≤ 𝑅/4, we have that ∥𝜃 𝑡 − 𝜃 ∗ ∥ P ≤ 𝑅/4 as well, meaning all iterates remain in 𝐵 𝑅/2 by induction. Then, recurring (34) within itself,   𝑇 −1 2 𝜅 2 2 𝜅 𝜅 𝑅 𝜅 ∑︁ ∥𝜃 𝑇 − 𝜃 ∗ ∥ P ≤ ∥𝜃 𝑇 −1 − 𝜃 ∗ ∥ P + ≤ ∥𝜃 𝑡 −2 − 𝜃 ∗ ∥ P + + ≤ · · · ≤ (2/3) 𝑇 + (2/3) 𝑗 3 3 3 3 3 3 4 3 𝑗=0 ≤ 2𝜅

(35)

Then, by the fundamental theorem of calculus and (23), since ∇ 𝑓𝜌 (𝜃 ∗ ) = 0, ∫ 1 𝑓𝜌 (𝜃 𝑇 ) − 𝑓𝜌 (𝜃 ∗ ) = 0

≤

(1 − 𝑡)𝐷 2 𝑓𝜌 ((1 − 𝑡)𝜃 ∗ + 𝑡𝜃 𝑇 ) [𝜃 𝑇 − 𝜃 ∗ , 𝜃 𝑇 − 𝜃 ∗ ]𝑑𝑡 ≤ 3∥𝜃 𝑇 − 𝜃 ∗ ∥ 2P

∫ 1 (1 − 𝑡)𝑑𝑡 0

3 ∥𝜃 𝑇 − 𝜃 ∗ ∥ 2P 2

Applying (35), this gives 𝑓𝜌 (𝜃 𝑇 ) − 𝑓𝜌 (𝜃 ∗ ) ≤ 6𝜅 2 ≤ 𝜖 which combines with (31) to prove the main guarantee. By Lemma 3.13, each gradient estimate uses 𝑂 (𝑛6 𝜅 −4 log(𝑛/𝛿)) copies by Lemma 3.13. Substituting the values for 𝜅, 𝑅, 𝑇, this reproduces the claimed copy complexity. □ Proposition 3.14 is already poly-time and its degraded 𝑂˜ (𝑛6 𝜖 −2 ) scaling is a technical artifact from using a zeroth-order method for gradient estimation.

23

3.4.4

Direct gradient estimation

We show that this artifact can be removed by directly estimating the gradient with a more complicated non-Gaussian measurement, achieving optimal copy complexity 𝑂˜ (𝑛2 𝜖 −1 ). Proposition 3.15. Suppose 𝑑tr (𝜌, |0⟩⟨0|) ≤ 𝛾0 and let opt ≡ inf | 𝜓⟩ ∈ G 𝑑IF (𝜌, |𝜓⟩⟨𝜓|). Then, given 𝜖 ∈ (0, 1) and 𝛿 ∈ (0, 1/2), there exists an algorithm which takes 𝑂˜ (𝑛2 𝜖 −1 log(1/𝛿)) copies of 𝜌 and outputs a pure ˆ such that Gaussian state | 𝜓⟩ ˆ ≤ opt + 𝜖 ˆ 𝜓|) 𝑑IF (𝜌, | 𝜓⟩⟨ with probability at least 1 − 𝛿. We begin by giving a direct gradient estimation protocol for the gradient at the vacuum state. Lemma 3.16 (Direct gradient estimation). For state 𝜎, 𝜅 ∈ (0, 1), and 𝛿 ∈ (0, 1/2), we can estimate ∇ 𝑓 𝜎 (0) to error 𝜅 in ∥ · ∥ P using 𝑂 (𝑛2 𝜅 −2 log(2/𝛿)) copies and general single-copy measurements. Proof. Let Π be the projector onto F≤2 , and define 𝑃 = Π ≤2 − |0⟩⟨0| and 𝑑 ≤2 = dim F≤2 = 1 + 𝑛 + 𝑛(𝑛 + 1)/2. Then, from the proof of Proposition 3.9, ⟨ℎ, ∇ 𝑓 𝜎 (0)⟩ P = 𝐷 𝑓 𝜎 (0) [ℎ] = −2 Re ⟨0| 𝐾 ℎ† 𝜎 |0⟩ = Re⟨𝐾 ℎ |0⟩ , −2𝑃𝜎 |0⟩⟩ where we use the fact that 𝐾 ℎ |0⟩ ∈ F≤2 is orthogonal to |0⟩ giving 𝑃𝐾 ℎ |0⟩ = 𝐾 ℎ |0⟩. Since we know that ⟨ℎ, 𝑔⟩ P = Re⟨𝐾 ℎ |0⟩ , 𝐾𝑔 |0⟩⟩, it follows that 𝐾 ∇ 𝑓 𝜎 (0) |0⟩ = −2𝑃𝜎 |0⟩ and that it suffices to estimate this quantity to ℓ2 error 𝑂 (𝜅), since we can simply read off the Bargmann parameters from the Fock coefficients of the state. We do this using the POVM M = {𝐼 − Π, 𝑑 ≤2 |𝑣⟩⟨𝑣| 𝑑𝜇(𝑣)} where 𝜇 is the uniform measure on the unit sphere of F≤2 . For outcome 𝑣, we record 𝑋 = (𝑑 ≤2 + 1)𝑃 |𝑣⟩ ⟨𝑣|0⟩, and for 𝐼 − Π, we record 𝑋 = 0. Then, ∫ E[𝑋] = 𝑑 ≤2 (𝑑 ≤2 + 1) 𝑃 |𝑣⟩ ⟨𝑣|0⟩ · ⟨𝑣| 𝜎 |𝑣⟩ 𝑑𝜇(𝑣)   ∫ ⟨𝑣| 𝜎 |𝑣⟩ |𝑣⟩⟨𝑣| 𝑑𝜇(𝑣) |0⟩ = (𝑑 ≤2 + 1)𝑃 𝑑 ≤2 = (𝑑 ≤2 + 1)𝑃 ·

Π𝜎Π + tr(Π𝜎)Π · |0⟩ 𝑑 ≤2 + 1

= 𝑃𝜎 |0⟩ where we use the standard Haar second-moment identity and use 𝑃 to annihilate the second term. For outcome 𝑣, ∥ 𝑋 ∥ 22 ≤ (𝑑 ≤2 + 1) 2 |⟨𝑣|0⟩| 2 since ∥𝑃 |𝑣⟩ ∥ 22 ≤ 1. Then, E[∥ 𝑋 ∥ 22 ] ≤ 𝑑 ≤2 (𝑑 ≤2 + 1) 2

∫

  ∫ ⟨𝑣| 𝜎 |𝑣⟩ |𝑣⟩⟨𝑣| 𝑑𝜇(𝑣) |0⟩ |⟨𝑣|0⟩| 2 ⟨𝑣| 𝜎 |𝑣⟩ 𝑑𝜇(𝑣) ≤ (𝑑 ≤2 + 1) 2 ⟨0| 𝑑 ≤2

Π𝜎Π + tr(Π𝜎)Π · |0⟩ 𝑑 ≤2 + 1 ≤ (𝑑 ≤2 + 1) (⟨0| 𝜎 |0⟩ + tr(Π𝜎)) ≤ (𝑑 ≤2 + 1) 2 ⟨0| ·

≤ 2(𝑑 ≤2 + 1) 24

A standard median-of-means estimator obtains error 𝜅 with 𝑂 (𝑑 ≤2 𝜅 −2 log(1/𝛿)) copies.

□

Since Lemma 3.16 provides an estimator for the gradient at vacuum, we can perform gradient descent in the same fashion as Proposition 3.14 except that we conjugate by a Gaussian unitary to send our current iterate to vacuum, then estimate the gradient, then take the step, and finally undo all the Gaussian unitaries we’ve accumulated. A similar analysis to the proof of Proposition 3.14 with this estimator then gives Theorem 3.8, using the smoothness and strong convexity to handle these additional rotations.

4

Agnostic learning in the low-fidelity regime

In this section we provide our agnostic learner for the low-fidelity regime. As mentioned above, the algorithm will work in two parts: first, we give an iterative unsqueezing procedure based on repeated calls to list-decodable learning that yields a short list of candidate bases, with the guarantee that the best Gaussian state approximator is approximately contained in the low photon number Fock subspace in one of these rotated bases. Then, we will show how to use tensor optimization to find this Gaussian approximator in this candidate basis. A Gaussian unitary 𝐺 defines the rotated basis {𝐺 † |𝑚⟩} of Fock space [BMEM25], which we identify with 𝐺; written in this basis, a state 𝜎 becomes 𝐺𝜎𝐺 † . Let 𝜋∗ be a closest pure Gaussian state to 𝜌, let 𝑉∗ denote its covariance, and let 𝐸 0 be an upper bound on its mean photon number 𝜈(𝜋∗ ), defined in Section 2. Throughout this section, we will assume that 𝐹 (𝜌, 𝜋∗ ) ≥ 𝛼. Throughout this section, we will often work in different Gaussian bases. Thus, for conciseness, when we say we perform a measurement in a basis 𝐺, we mean applying 𝐺 to fresh copies of 𝜌 before measuring them. Note that such a change of basis preserves fidelity, but can change the mean photon number. The unrotated Fock basis is 𝐺 = 𝐼, and a state obtained in the basis 𝐺 is rotated back by applying 𝐺 † .

4.1

List-decodable iterative unsqueezing

This subsection gives the algorithm for constructing a short list of rotated bases, in one of which 𝜋∗ has low photon number. Notation.

For a rotated basis 𝐺 and a state 𝜎, let 𝑁𝐺 = 𝐺 † 𝑁𝐺,

𝜈𝐺 (𝜎) = tr(𝜎𝑁𝐺 ) = 𝜈(𝐺𝜎𝐺 † )

be the photon-number operator and the mean photon number in the basis 𝐺. When we are working in a basis 𝐺, we write m𝐺 and 𝑉𝐺 for the mean and covariance of 𝐺𝜋∗ 𝐺 † , and measure its squeezing by 𝜅(𝑉) = ∥𝑉 ∥ op = ∥𝑉 −1 ∥ op . When the basis is clear, we will often drop the subscripts. We call 𝐺 useful at level 𝐴 if 𝜅(𝑉) ≤ 𝐴. Heterodyne measurement of 𝐺𝜋∗ 𝐺 † produces samples from N (m, Σ), where Σ = (𝑉 + 𝐼)/2 is its heterodyne covariance. For a symplectic matrix 𝑆, we write 𝑈𝑆 for the Gaussian unitary with no displacement that maps a state with moments (m, 𝑉) to one with moments (𝑆m, 𝑆𝑉 𝑆 ⊤ ). Finally, let 𝑡 𝛼 = 𝐶dec (4/𝛼) 𝑟dec , where 𝐶dec , 𝑟 dec > 1 are the universal constants of Lemma 2.10. The main output of this section is the following theorem: 25

Theorem 4.1 (List-decodable unsqueezing). Fix 𝛼 ∈ (0, 1], 𝐴0 ≥ 1, and 𝜂 ∈ (0, 1/2]. There is an algorithm which, given copies of an arbitrary 𝑛-mode state 𝜌, returns a list of at most 𝑊 = ⌈8/(3𝛼)⌉ Gaussian bases, or reports failure. For every fixed best Gaussian approximator 𝜋∗ satisfying 𝐹 (𝜌, 𝜋∗ ) ≥ 𝛼,

𝜅(𝑉∗ ) ≤ 𝐴0 ,

with probability at least 1 − 𝜂 it returns a list containing a basis 𝐺 such that 𝜈(𝐺𝜋∗ 𝐺 † ) ≤ 𝐻 𝛼 ,

(36)

where we may take 𝐻 𝛼 = 𝑒 K2𝛼 ≍ 𝛼 −𝑠 for a universal exponent 𝑠 ≍ 𝑟 dec , with K 𝛼 defined in (49). The number of copies and total runtime of the algorithm are both at most
exp(𝐶𝑡 𝛼 )𝑛𝑡 𝛼 poly log(1/𝜂), log log(𝑒 + 𝐴0 ) ,

(37)

where 𝐶dec , 𝑟 dec are as in Lemma 2.10. The algorithm runs in blocks. Each block consists of 𝑞 rounds, followed by a fit and a pruning step. In each round, every basis in the current list is replaced by its updated bases, one for each candidate in a list estimate (Section 4.1.1). Along a branch that always follows an accurate candidate, each round roughly takes the square root of the squeezing of 𝜋∗ (Section 4.1.2). After 𝑞 rounds, a final fit changes to bases in which estimates of 𝜋∗ become the vacuum; this controls the photon number of 𝜋∗ , not only its squeezing. The block then prunes the list back to at most 𝑊 bases, using a photon-cutoff test and pairwise comparisons of bases (Section 4.1.3). Section 4.1.4 states the algorithm and proves Theorem 4.1. 4.1.1

List estimates

Heterodyne list estimates. We first record what one run of list-decodable estimation guarantees in a single basis. The next lemma applies Lemma 2.10 to heterodyne outcomes. Lemma 4.2 (List estimates). In any adaptively chosen basis, a classical list decoding algorithm run on 𝑠 = 𝑂 (𝑒 𝑡 𝛼 (3𝑛) 𝑡 𝛼 ) fresh heterodyne outcomes returns at most 𝑂 (𝑒 𝑡 𝛼 ) pairs, each with positive definite covariance. With probability at least 9/10, one pair ( 𝜇, ˆ Σ̂) satisfies all three bounds 𝑡 −1 𝛼 Σ ⪯ Σ̂ ⪯ 𝑡 𝛼 Σ,

Σ −1/2 ( Σ̂ − Σ)Σ −1/2 𝐹 ≤ 𝑡 2𝛼 , √︁ Σ −1/2 ( 𝜇ˆ − m) 2 ≤ 𝑡 𝛼 (1 + 𝑡 𝛼 ).

(38)

Proof. Let 𝑃 and 𝑄 = N (m, Σ) be the heterodyne laws of the input and 𝜋∗ in the chosen basis. Measurement contractivity and the pure-state fidelity bound imply √ 1 − 𝑑tv (𝑃, 𝑄) ≥ 1 − 𝑑tr (𝜌, 𝜋) ≥ 1 − 1 − 𝛼 ≥ 𝛼/2. The result then directly follows from Lemma 2.10.

□

Good pairs. We call a pair ( 𝜇, ˆ Σ̂) good for a basis 𝐺 if it satisfies the three bounds (38), where m and Σ are the mean and heterodyne covariance of 𝜋∗ in the basis 𝐺. By Lemma 4.2, a single list contains a good 26

pair with probability at least 9/10. To boost this, the algorithm runs the decoder 𝑏 times on fresh copies and takes the union of the resulting lists, and so it fails to contain a good pair with probability at most 10−𝑏 . Updated bases. Each pair ( 𝜇, ˆ Σ̂) in a list determines a new basis. Inflate the covariance estimate and take a Williamson decomposition, b ⊤, 𝑉ˆ = 2ΛΣ̂ = 𝑆 𝐷𝑆

𝑉 + 𝐼 ⪯ 𝑉ˆ ⪯ Λ2 (𝑉 + 𝐼),

(39)

b = diag(𝑑1 , . . . , 𝑑 𝑛 , 𝑑1 , . . . , 𝑑 𝑛 ) with 𝑑 𝑗 > 0, and the inequalities hold whenever the where 𝑆 is symplectic, 𝐷 pair is good, by the spectral bound in (38). If the current basis is 𝐺, the updated basis of the pair is 𝑈𝑆 −1 𝐺. b In the updated basis, a covariance 𝑉 becomes 𝑆 −1𝑉 𝑆 −⊤ ; in particular, 𝑉ˆ becomes the diagonal matrix 𝐷. The inflation by 2Λ makes 𝑉ˆ an overestimate of 𝑉, which is what the unsqueezing lemma requires. 4.1.2

Unsqueezing and the fit

The unsqueezing step. We now show that changing to the updated basis of a good pair roughly takes the square root of the squeezing of 𝜋∗ . Let 𝑉 be the covariance of 𝜋∗ in the current basis, and let 𝜅 = 𝜅(𝑉) = ∥𝑉 −1 ∥ op ≥ 1 be its squeezing. For a good pair, (39) shows that 𝑉ˆ overestimates 𝑉 by at most a constant multiple of 𝑉 + 𝐼. Indeed, 𝜅𝑉 ⪰ 𝐼 gives 𝑉 ⪰ (𝑉 + 𝐼)/(𝜅 + 1), so  0 ⪯ 𝑉ˆ − 𝑉 ⪯ Λ2 −

1  (𝑉 + 𝐼). 𝜅+1

The coefficient on the right is positive, so this is the hypothesis (12) of Lemma 2.8 with 𝜁 ∈ (0, 1) chosen so that 2𝜁/(1 − 𝜁) = Λ2 − 1/(𝜅 + 1). The algorithm never needs 𝜁, which depends on the unknown 𝑉; the updated basis is computed from 𝑉ˆ alone. The lemma bounds the squeezing of 𝜋∗ in the updated basis, where its covariance is 𝑉 ′ = 𝑆 −1𝑉 𝑆 −⊤ : √︂  √ √ √ 1  ∥ (𝑉 ′ ) −1 ∥ op ≤ 1 + Λ2 − (𝜅 + 1) = Λ 𝜅 + 1 ≤ 2 Λ 𝜅, 𝜅+1 √ √ where the last step uses 𝜅 ≥ 1. Thus one round replaces the squeezing 𝜅 by at most 2Λ 𝜅. Now follow good pairs for 𝑞 consecutive√rounds, and let 𝜅0 , 𝜅1 , . . . , 𝜅 𝑞 be the squeezing of 𝜋∗ in the √ successive bases, with 𝜅0 ≤ 𝐴. Since 𝜅 𝑗+1 ≤ 2Λ 𝜅 𝑗 for each 𝑗, unrolling the recursion gives √ √ 1 − (𝑞−1) −𝑞 −𝑞 −𝑞 −𝑞 𝜅 𝑞 ≤ ( 2Λ) 1+ 2 +···+2 𝐴2 = ( 2Λ) 2(1−2 ) 𝐴2 ≤ 2Λ2 𝐴2 ,

(40)

√ where the last step uses 2Λ ≥ 1. The constant factors do not accumulate: their exponents form a geometric series that sums to less than 2. So however many rounds we take, the overhead is at most 2Λ2 , while the exponent of 𝐴 shrinks by a factor of 2𝑞 . The final output. After 𝑞 rounds, the squeezing of 𝜋∗ is bounded independently of 𝑛 along a good branch. This alone does not bound its photon number independently of 𝑛: a product of 𝑛 single-mode squeezed states, each with squeezing 𝐴, has photon number 𝑛( 𝐴 + 𝐴 −1 − 2)/4. To remove the dependence on 𝑛, we make one final change of basis, to a basis in which an estimate of 𝜋∗ becomes the vacuum. Its analysis uses the Frobenius and mean bounds in (38), which control the total estimation error across all modes, and not 27

only the spectral bound, which controls the error in each direction separately. For the same reason, the fit ˆ inflating by a factor of Λ adds a relative starts from the raw estimate Σ̂ rather than from the inflated matrix 𝑉: √ error close to (Λ − 1)𝐼, whose Frobenius norm grows like 𝑛. The estimate Σ̂ need not be the heterodyne covariance of a pure state, so we first round it to one. This is easiest in terms of the heterodyne precision 𝐾 = Σ −1 = 2(𝑉 + 𝐼) −1 , because purity is a linear constraint on 𝐾. With Ω = Ω𝑛 , a matrix 0 ≺ 𝐾 ≺ 2𝐼 is the precision of a pure Gaussian state if and only if 𝐾 + Ω𝐾Ω⊤ = 2𝐼.

(41)

Indeed, if 𝑉 is pure then Ω𝑉Ω⊤ = 𝑉 −1 , so Ω𝐾Ω⊤ = 2(𝑉 −1 + 𝐼) −1 = 2𝑉 (𝑉 + 𝐼) −1 , and adding 𝐾 = 2(𝑉 + 𝐼) −1 gives 2𝐼. Conversely, if 𝐾 satisfies (41), then 𝑉 = 2𝐾 −1 − 𝐼 is positive definite and Ω𝑉Ω⊤ = 2(2𝐼 −𝐾) −1 − 𝐼 = 𝐾 (2𝐼 − 𝐾) −1 = 𝑉 −1 , so 𝑉 is the covariance of a pure state. We also keep the eigenvalue bounds that the true precision satisfies. If 𝐴 −1 𝐼 ⪯ 𝑉 ⪯ 𝐴𝐼, then the eigenvalues of 𝐾 lie in [2/( 𝐴 + 1), 2𝐴/( 𝐴 + 1)], so 𝐾 lies in the set n
o 1 1 K 𝐴 = 𝐾 = 𝐾 ⊤ : 𝐾 + Ω𝐾Ω⊤ = 2𝐼, 𝐴+1 𝐼 ⪯ 𝐾 ⪯ 2 − 𝐴+1 𝐼 . (42) This set is closed, convex and nonempty, since it contains the precision 𝐼 of the vacuum. So every symmetric matrix 𝐾ˆ has a unique closest point in K 𝐴 in Frobenius norm, and it has a closed form. First project onto ˆ ⊤ )/2; then clip the eigenvalues of 𝑋 to [− 𝐴 , 𝐴 ]. the affine constraint, 𝐾ˆ ↦→ 𝐼 + 𝑋 with 𝑋 = ( 𝐾ˆ − Ω𝐾Ω 𝐴+1 𝐴+1 Clipping preserves the affine constraint, because Ω𝑋Ω⊤ = −𝑋 and clipping is an odd function of 𝑋. Since the first map is an orthogonal projection and clipping is the Frobenius projection onto the eigenvalue constraint, their composition is the Frobenius projection onto K 𝐴. The fit takes a terminal pair, meaning a pair ( 𝜇, ˆ Σ̂) from a list computed in the final basis 𝐺 of a block. −1 Let 𝐾0 be the point of K 𝐴 closest to Σ̂ . By the equivalence above, 𝑉0 = 2𝐾0−1 − 𝐼 is a pure covariance; write 𝑉0 = 𝑆0 𝑆0⊤ with 𝑆0 symplectic. The fitted basis is obtained from 𝐺 by the displacement by − 𝜇ˆ followed by 𝑈𝑆 −1 . In it, the pure Gaussian state with mean 𝜇ˆ and covariance 𝑉0 , which is our estimate of 𝜋∗ , becomes 0 the vacuum. The algorithm fits every terminal pair. In the next lemma, 𝑉 is the covariance of 𝜋∗ in the basis 𝐺 where the terminal list was computed. Lemma 4.3 (the fitted basis). If 𝐴 −1 𝐼 ⪯ 𝑉 ⪯ 𝐴𝐼 and the terminal pair satisfies (38), then the fitted basis maps the best Gaussian approximator to a state with total mean photon number at most ℎ 𝛼 ( 𝐴) = 𝑢 𝛼 𝐴8 ,

𝑢 𝛼 = 1200 𝑡 6𝛼 + 𝑡 𝛼 (1 + 𝑡 𝛼 ) ≍ 𝑡 6𝛼 .

We defer the proof to Appendix A.3.1. 4.1.3

Pruning the list

We prune the bases in two ways. First, in a useful basis 𝜌 has substantial weight at low photon number, so we discard bases in which 𝜌 has little weight below a photon cutoff. Second, among the remaining bases, we discard any basis that is close to one we keep. We measure the closeness of two bases 𝐺 and 𝐻 by Δ(𝐺, 𝐻) = 𝜆 min (𝑁𝐺 + 𝑁 𝐻 ),

28

the smallest possible total photon number 𝜈𝐺 (𝜎) + 𝜈 𝐻 (𝜎) of a state 𝜎 in the two bases. It is symmetric, and Δ(𝐺, 𝐺) = 0. The next lemma computes Δ from the relative Gaussian transformation between the two bases, without manipulating operators on the full Fock space. Lemma 4.4. Let 𝑈 = 𝐺𝐻 † , and let 𝑈 |0⟩ have mean d, covariance 𝐶, and squeezing parameters 𝑟 1 , . . . , 𝑟 𝑛 ≥ 0. Then 𝑛 ∑︁ 1 (43) Δ(𝐺, 𝐻) = (cosh 𝑟 𝑗 − 1) + d⊤ (𝐼 + 𝐶) −1 d. 2 𝑗=1 Thus Δ is computable from the relative Gaussian transformation using polynomially many matrix operations. We defer the proof to Appendix A.3.2. Low-photon subspaces. For an integer 𝑘 ≥ 1, let Π ≤ 𝑘 be the projection onto states with total photon number at most 𝑘, and let 𝑃𝐺 (𝑘) = 𝐺 † Π ≤ 𝑘 𝐺 be the corresponding projection in the basis 𝐺. We call tr(𝜌𝑃𝐺 (𝑘)) the cutoff mass of 𝜌 in the basis 𝐺. Applying 𝐺 to a fresh copy of 𝜌 and testing whether the photon number is at most 𝑘 succeeds with probability exactly tr(𝜌𝑃𝐺 (𝑘)). By Hoeffding’s inequality, 𝑂 (𝑎 −2 log(1/𝜏)) copies therefore estimate the cutoff mass to additive error 𝑎 with failure probability at most 𝜏, independently of 𝑘. The next proposition gives the two properties of Δ that make pruning work. Part (i) says that if Δ(𝐺, 𝐻) is small, then a pure Gaussian state with few photons in one of the bases also has few photons in the other, so discarding one of the two bases loses little. Part (ii) says that if Δ(𝐺, 𝐻) is large, then the low-photon subspaces of 𝐺 and 𝐻 are nearly orthogonal. By the packing bound of Lemma 4.6 below, 𝜌 can then have large cutoff mass in only a few bases that are pairwise far apart. So we fix a threshold 𝑇 and discard a basis whenever Δ between it and some kept basis is at most 𝑇: by (i) this costs only a bounded factor in the photon number, and by (ii) few bases survive. Proposition 4.5. Let constants 𝐶tr = 32, 𝑑 = 32, and a sufficiently large universal 𝐶sep ≥ 1. (i) For every pure Gaussian 𝜋 with 𝜈𝐺 (𝜋) ≤ ℎ and ℎ ≥ 1, 𝜈 𝐻 (𝜋) ≤ 𝐶tr (Δ(𝐺, 𝐻) + 1) 3 ℎ. (ii) For every integer 𝑘 ≥ 1 and 0 < 𝜇 ≤ 1/2, set  𝑑 𝑘 𝑇 (𝑘, 𝜇) = 𝐶sep . 𝜇 If Δ(𝐺, 𝐻) > 𝑇 (𝑘, 𝜇), then ∥𝑃𝐺 (𝑘)𝑃 𝐻 (𝑘) ∥ op ≤ 𝜇. Proof of (i). Work in basis 𝐺. Let (m, 𝑉) be the moments of 𝜋∗ and (d, 𝐶) those of the 𝐻-vacuum in this basis. Writing 𝐶 = 𝑆𝑆 ⊤ , the passage to basis 𝐻 transforms these moments to 𝑆 −1 (m − d) and 𝑆 −1𝑉 𝑆 −⊤ , up to a passive transformation. Hence
tr 𝐶 −1𝑉 − 2𝑛 1 𝜈 𝐻 (𝜋) = + (m − d) ⊤𝐶 −1 (m − d). (44) 4 2 29

Put Δ = Δ(𝐺, 𝐻) and 𝑏 = (tr 𝐶 − 2𝑛)/4. By (43), ∑︁ ∑︁ ∑︁ 𝑏= sinh2 𝑟 𝑗 = (cosh 𝑟 𝑗 − 1) 2 + 2 (cosh 𝑟 𝑗 − 1) ≤ Δ(Δ + 2), 𝑗 ±1

𝑗 2

∥𝐶 ∥ op ≤ 4(Δ + 1) ,

𝑗 ⊤

−1

𝑄 := d (𝐼 + 𝐶) d ≤ 2Δ.

Comparing eigenvalues of 𝐶 −1 and (𝐼 + 𝐶) −1 then gives d⊤𝐶 −1 d ≤ (1 + ∥𝐶 −1 ∥ op )𝑄 ≤ 10(Δ + 1) 3 . To control the trace term without introducing a factor of 𝑛, note that a pure covariance 𝑊 with 𝑢 = (tr 𝑊 − 2𝑛)/4 satisfies ∥𝑊 − 𝐼 ∥ 2𝐹 ≤ 16𝑢 2 + 8𝑢 ≤ 16(𝑢 + 1) 2 . −1 2 Indeed, its reciprocal eigenvalue pair (𝜆 𝑗 , 𝜆 −1 𝑗 ) contributes 16𝑢 𝑗 + 8𝑢 𝑗 , where 𝑢 𝑗 = (𝜆 𝑗 + 𝜆 𝑗 − 2)/4 ≥ 0 and Í −1 − 𝐼 and 𝐵 = 𝑉 − 𝐼, purity and ℎ ≥ 1 give tr 𝐴 = 4𝑏, tr 𝐵 ≤ 4ℎ, ∥ 𝐴∥ ≤ 4(𝑏 + 1), 𝐹 𝑗 𝑢 𝑗 = 𝑢. For 𝐴 = 𝐶 and ∥𝐵∥ 𝐹 ≤ 5ℎ. Therefore   tr 𝐶 −1𝑉 − 2𝑛 = tr 𝐴 + tr 𝐵 + tr( 𝐴𝐵) ≤ 4𝑏 + 4ℎ + 20(𝑏 + 1)ℎ ≤ 28(Δ + 1) 2 ℎ.

Finally ∥m∥ 22 ≤ 2ℎ, so (m − d) ⊤𝐶 −1 (m − d) ≤ 2∥𝐶 −1 ∥ op ∥m∥ 22 + 2d⊤𝐶 −1 d ≤ 36(Δ + 1) 3 ℎ. Substitution into (44) gives 𝜈 𝐻 (𝜋) ≤ 25(Δ + 1) 3 ℎ ≤ 𝐶tr (Δ + 1) 3 ℎ, proving (i).

□

Proof of (ii). With 𝑈 = 𝐺𝐻 † , unitary invariance gives ∥𝑃𝐺 (𝑘)𝑃 𝐻 (𝑘) ∥ op = ∥Π ≤ 𝑘 𝑈Π ≤ 𝑘 ∥ op . Suppose this is greater than 𝜇, and choose unit vectors 𝜓, 𝜒 ∈ Im(Π ≤ 𝑘 ) with |⟨𝜓, 𝑈 𝜒⟩| > 𝜇. Write (d, 𝐶) for the moments of 𝑈 |0⟩ and 𝑟 𝑗 for its squeezes. We first bound the largest squeeze, then use the definition of Δ(𝐺, 𝐻) to control the total compatibility cost. For every real vector 𝑥 and every unit vector 𝜑 ∈ Im(Π ≤ 𝑘 ), ⟨𝜑, (𝑥 ⊤ R̂) 2 𝜑⟩ ≲ 𝑘 ∥𝑥∥ 22 . √ Indeed, a passive transformation sends 𝑥 ⊤ R̂ to ∥𝑥∥ 2 𝑞ˆ 1 and preserves the cutoff; use 𝑞ˆ 1 = ( 𝑎ˆ 1 + 𝑎ˆ †1 )/ 2 √ and ∥ 𝑞ˆ 1 Π ≤ 𝑘 ∥ op ≲ 𝑘, since 𝑘√≥ 1. We also need the following density bound: every unit quadrature has probability density at most 𝐶 𝐻 𝑘 on a cutoff-𝑘 state, for a universal 𝐶 𝐻 . After a passive rotation and partial Í trace, its density is at most 𝑘𝑗=0 |ℎ 𝑗 (𝑡)| 2 , where ℎ 𝑗 are normalized Hermite functions. Mehler’s identity gives, for 0 < 𝑣 < 1,   𝑘 ∑︁ 1−𝑣 2 𝑣 −𝑘 exp − 𝑡 . |ℎ 𝑗 (𝑡)| 2 ≤ √︁ 1+𝑣 𝜋(1 − 𝑣 2 ) 𝑗=0 Taking 𝑣 = 𝑒 −1/𝑘 proves the bound. Scaling the quadrature by 𝑠 divides the density bound by 𝑠, and 30

translation leaves it unchanged. Let 𝑠 = 𝑒𝑟max and ∫measure a unit quadrature corresponding to the largest eigenvalue 𝑠2 of 𝐶. Its densities √ √ 2 𝑝, 𝑞 in 𝜓, 𝑈 𝜒 satisfy 𝑡 𝑝(𝑡) 𝑑𝑡 ≲ 𝑘 and ∥𝑞∥ ∞ ≲ 𝑘/𝑠. Take 𝑅 = 𝐶0 𝑘/𝜇 for a sufficiently large universal 𝐶0 , and let 𝐼 𝑅 = [−𝑅, 𝑅]. Then 𝑝(𝐼 𝑅𝑐 ) ≤ 𝑐𝜇2 ,

𝑞(𝐼 𝑅 ) ≲ 𝑘/(𝜇𝑠),

where 𝑐 is a sufficiently small universal constant. Fidelity monotonicity under this measurement and Cauchy–Schwarz give ∫ √︃ √︁ √ 𝜇 < |⟨𝜓, 𝑈 𝜒⟩| ≤ 𝑝𝑞 ≤ 𝑞(𝐼 𝑅 ) + 𝑝(𝐼 𝑅𝑐 ). For 𝑠 larger than a sufficiently large universal multiple of 𝑘 𝜇 −3 , the right-hand side is at most 𝜇, a contradiction. Thus 𝑒𝑟max ≲ 𝑘 𝜇 −3 . (45) Now set 𝑃 = Π ≤ 𝑘 , 𝐵 = 𝑈𝑁𝑈 † , 𝑄 = 𝑈𝑃𝑈 † , and Δ = Δ(𝐺, 𝐻). By the original definition, 𝑁 + 𝐵 ⪰ Δ𝐼. Since 𝑃, 𝑄 have finite rank, there are unit vectors 𝜙 ∈ Im 𝑃, 𝜂 ∈ Im 𝑄 such that 𝑃𝜂 = 𝑡𝜙,

𝑄𝜙 = 𝑡𝜂,

𝑡 = ∥𝑃𝑄∥ op = ∥𝑃𝑈𝑃∥ op > 𝜇.

The operator 𝐵 commutes with 𝑄, so ⟨𝜙, 𝐵𝜂⟩ = 𝑡⟨𝜂, 𝐵𝜂⟩. Expanding 𝜂 = 𝑡𝜙 + (𝐼 − 𝑃)𝜂 therefore gives 𝑡⟨𝜙, 𝐵𝜙⟩ = 𝑡⟨𝜂, 𝐵𝜂⟩ − ⟨(𝐼 − 𝑃)𝐵𝜙, (𝐼 − 𝑃)𝜂⟩. Since ⟨𝜙, 𝑁 𝜙⟩ ≤ 𝑘 and ⟨𝜂, 𝐵𝜂⟩ ≤ 𝑘, we obtain Δ ≤ ⟨𝜙, (𝑁 + 𝐵)𝜙⟩ ≤ 2𝑘 +

∥ (𝐼 − 𝑃)𝐵𝑃∥ op ∥ (𝐼 − 𝑃)𝐵𝑃∥ op ≤ 2𝑘 + . 𝑡 𝜇

(46)

Thus it suffices to bound the part of 𝐵 that crosses the photon cutoff. Write the quadratic Hamiltonian as 𝐵 = 𝑎ˆ † 𝑀 𝑎ˆ +

1 †⊤ † 𝑎ˆ 𝛽 𝑎ˆ + 𝑎ˆ ⊤ 𝛽 𝑎ˆ + 𝑎ˆ † ℓ + ℓ † 𝑎ˆ + 𝑐 0 𝐼, 2

𝑀 = 𝑀 †,

𝛽 = 𝛽⊤ .

Only the terms creating one or two photons contribute to (𝐼 − 𝑃)𝐵𝑃. For 𝑣 supported on at most 𝑘 + 2 photons, Cauchy–Schwarz gives ! 1/2 ∑︁ 1 ⊤ 1 𝑎ˆ 𝛽 𝑎ˆ 𝑣 ≤ ∥ 𝛽∥ 𝐹 ∥ 𝑎ˆ 𝑖 𝑎ˆ 𝑗 𝑣∥ 22 ≲ 𝑘 ∥ 𝛽∥ 𝐹 ∥𝑣∥ 2 , 2 2 2 𝑖, 𝑗 where the sum equals ⟨𝑣, 𝑁 (𝑁 − 1)𝑣⟩ and √ 𝑘 ≥ 1. Taking adjoints bounds the pair-creation term on Im 𝑃; the one-photon term satisfies ∥ 𝑎ˆ † ℓ𝑃∥ op ≲ 𝑘 ∥ℓ∥ 2 . Consequently, √ ∥ (𝐼 − 𝑃)𝐵𝑃∥ op ≲ 𝑘 ∥ 𝛽∥ 𝐹 + 𝑘 ∥ℓ∥ 2 .

31

(47)

Expanding 𝐵 = 12 ( R̂ − d) ⊤𝐶 −1 ( R̂ − d) − 𝑛𝐼/2 in ladder operators gives ∥ 𝛽∥ 2𝐹 =

∑︁

1 ∥ℓ∥ 22 = d⊤𝐶 −2 d. 2

sinh2 (2𝑟 𝑗 ),

𝑗

For the first identity, a passive change of modes diagonalizes 𝐶; the pair coefficient in mode 𝑗 then has
magnitude sinh 2𝑟 𝑗 . The second follows from the linear term −d⊤𝐶 −1 R̂. By (43) and 𝑠 −2 𝐼 ⪯ 𝐶 ⪯ 𝑠2 𝐼, ∑︁ ∥ 𝛽∥ 2𝐹 = 4 (cosh 𝑟 𝑗 − 1) (cosh 𝑟 𝑗 + 1) cosh2 𝑟 𝑗 ≲ 𝑠3 Δ, 𝑗

∥ℓ∥ 22 ≤

𝑠4 + 𝑠2 2

d⊤ (𝐼 + 𝐶) −1 d ≲ 𝑠4 Δ.

Substituting into (47) and then (46) yields Δ ≤ 2𝑘 +

√
√ 𝐶∗ 𝑘 𝑠3/2 + 𝑘 𝑠2 Δ 𝜇

for a universal constant 𝐶∗ . Young’s inequality and (45) now give Δ ≲ 𝑘 + 𝜇 −2 (𝑘 2 𝑠3 + 𝑘 𝑠4 ) ≲ 𝑘 5 𝜇 −14 ≤ (𝑘/𝜇) 32 . Choosing 𝐶sep sufficiently large proves (ii) by contradiction.

□

Part (ii) limits how many bases survive pruning through the following packing bound. Lemma 4.6. Let 𝑃1 , . . . , 𝑃𝑠 be orthogonal projections satisfying tr(𝜌𝑃𝑖 ) ≥ 𝑤 and ∥𝑃𝑖 𝑃 𝑗 ∥ op ≤ 𝜇 for 𝑖 ≠ 𝑗, where 𝑤 > 𝜇. Then 1−𝜇 𝑠≤ . 𝑤−𝜇 Proof. For 𝑥 𝑖 ∈ Im(𝑃𝑖 ), |⟨𝑥𝑖 , 𝑥 𝑗 ⟩| ≤ 𝜇∥𝑥𝑖 ∥ 2 ∥𝑥 𝑗 ∥ 2 , so 2

∑︁

≤ (1 + (𝑠 − 1)𝜇)

𝑥𝑖 2

𝑖

The map (𝑥1 , . . . , 𝑥 𝑠 ) ↦→ Í 𝑖 𝑃𝑖 , giving

Í

∑︁

∥𝑥𝑖 ∥ 22 .

𝑖

𝑖 𝑥 𝑖 thus has squared norm at most 1 + (𝑠 − 1)𝜇. Its product with its adjoint is

! 𝑠𝑤 ≤ tr 𝜌

∑︁

𝑃𝑖 ≤

∑︁ 𝑖

𝑖

≤ 1 + (𝑠 − 1)𝜇.

𝑃𝑖 op

This proves the lemma.

□

Finally, in a basis where 𝜋∗ has bounded photons, 𝜌 has large cutoff mass, so useful bases survive the mass test. Lemma 4.7. If 𝐹 (𝜌, 𝜋∗ ) ≥ 𝛼 and 𝜈𝐺 (𝜋∗ ) ≤ ℎ for ℎ ≥ 1, then one may take 𝐶0 = 64 in the bound 𝑘 ≥ 𝐶0 ℎ/𝛼

=⇒

tr(𝜌𝑃𝐺 (𝑘)) ≥ 3𝛼/4. 32

Proof. Write 𝜋∗ = |𝜓⟩⟨𝜓| and 𝑃 = 𝑃𝐺 (𝑘). Markov’s inequality gives 𝜁 = ∥ (𝐼 − 𝑃)𝜓∥ 22 ≤ ℎ/𝑘 ≤ 𝑐𝛼, where 𝑐 = 𝐶0−1 = 1/64. Since 0 ⪯ 𝜌 ⪯ 𝐼, √

√︁ √︁ √ √ √ 𝛼 ≤ ∥ 𝜌𝜓∥ 2 ≤ ∥ 𝜌𝑃𝜓∥ 2 + ∥ 𝜌(𝐼 − 𝑃)𝜓∥ 2 ≤ tr(𝜌𝑃) + 𝜁 .

Here the squared first term is at most tr(𝑃𝜌𝑃), and the second is at most 𝜁. Consequently tr(𝜌𝑃) ≥ √ √ √ ( 𝛼 − 𝜁) 2 ≥ (1 − 𝑐) 2 𝛼 = 49𝛼/64 ≥ 3𝛼/4. □ 4.1.4

The algorithm and its analysis

We now put everything together. Parameters for one block. Set 𝑞 = 11 and 𝜇 = 𝛼/16. Recall the photon bound ℎ 𝛼 of Lemma 4.3 and the constants 𝐶tr , 𝑑 and 𝐶sep of Proposition 4.5. For a squeezing bound 𝐴 ≥ 1 at the start of a block, define −𝑞

𝐵( 𝐴) = 2Λ2 𝐴2 ,

ℎ( 𝐴) = ℎ 𝛼 (𝐵( 𝐴)), 𝑑  𝑘 ( 𝐴) , 𝑇 ( 𝐴) = 𝐶sep 𝜇

𝑘 ( 𝐴) = ⌈64ℎ( 𝐴)/𝛼⌉ , ℎ+ ( 𝐴) = 8𝐶tr𝑇 ( 𝐴) 3 ℎ( 𝐴),

𝐴+ ( 𝐴) = 6ℎ+ ( 𝐴).

For conciseness, throughout this section, we will refer to the list-deocdable learning algorithm in Lemma 4.2 as the decoder algorithm. For an integer 𝑏 ≥ 1, let List𝑏 (𝐺) be the union of 𝑏 independent runs of the decoder in the basis 𝐺, each on fresh copies. Write Fit 𝐴 (𝐺, 𝜇, ˆ Σ̂) for the fitted basis of the terminal pair ( 𝜇, ˆ Σ̂) in the basis 𝐺, computed with 𝐵( 𝐴) in place of 𝐴 in (42). Every cutoff-mass estimate in a block has additive error 𝛼/16 and failure probability 𝜏. With this, we can now give pseudocode (see Algorithm 1) for processing one block.

33

Algorithm 1: One block of list-decodable unsqueezing Input: Copies of 𝜌, 𝛼 ∈ (0, 1], a basis list L with |L| ≤ 𝑊, a bound 𝐴 ≥ 1, repetitions 𝑏 ≥ 1, and 𝜏 ∈ (0, 1/2]. Output: A list L + of at most 𝑊 bases, or failure. 1 Set T0 = L; 2 for 𝑗 = 0, . . . , 𝑞 − 1 do 3 Set T𝑗+1 = ∅; 4 foreach 𝐺 ∈ T𝑗 do 5 foreach ( 𝜇, ˆ Σ̂) ∈ List𝑏 (𝐺) do b ⊤; 6 Compute a Williamson decomposition 2𝑡 𝛼 Σ̂ = 𝑆 𝐷𝑆 7 Append 𝑈𝑆 −1 𝐺 to T𝑗+1 ; 8 Set C = ∅; 9 foreach 𝐺 ∈ T𝑞 do 10 11

foreach ( 𝜇, ˆ Σ̂) ∈ List𝑏 (𝐺) with Σ̂ ≻ 0 do Append Fit 𝐴 (𝐺, 𝜇, ˆ Σ̂) to C;

12 foreach 𝐻 ∈ C do 13

Estimate tr(𝜌𝑃 𝐻 (𝑘 ( 𝐴))) by 𝑤ˆ 𝐻 ;

14 Set L + = ∅; 15 foreach 𝐻 ∈ C with 𝑤 ˆ 𝐻 ≥ 𝛼/2, in a fixed order do 16 if Δ(𝐻, 𝐽) > 𝑇 ( 𝐴) for every 𝐽 ∈ L + then 17 18 19

Append 𝐻 to L + ; if |L + | > 𝑊 then Return failure;

20 Return L + ;

The next lemma shows that a block preserves the invariant: if the input list contains a basis useful at level 𝐴, then the output list contains a basis useful at level 𝐴+ ( 𝐴), and the list stays short. Lemma 4.8 (One block step). Suppose 𝐹 (𝜌, 𝜋∗ ) ≥ 𝛼 and the input list contains a basis in which the best Gaussian approximator 𝜋∗ has squeezing at most 𝐴. If the estimation lists contain good pairs and the cutoff-mass estimates in Algorithm 1 are accurate, the next list has at most 8/(3𝛼) bases and contains a basis 𝐽 with 𝜈 𝐽 (𝜋∗ ) ≤ ℎ+ ( 𝐴), 𝜅(𝑉𝐽 ) ≤ 𝐴+ ( 𝐴). Proof. Follow a good list element at each of the 𝑞 levels. Equation (40), obtained from Lemma 2.8, bounds 𝜋∗ squeezing at the resulting leaf by 𝐵( 𝐴). A good terminal pair and Lemma 4.3 then give a candidate 𝐻∗ with 𝜈 𝐻∗ (𝜋∗ ) ≤ ℎ( 𝐴). By Lemma 4.7, its true cutoff mass is at least 3𝛼/4, and its estimated mass is therefore at least 3𝛼/4 − 𝛼/16 > 𝛼/2. It survives the mass test. If retained, it already has the required photon bound. If discarded during merging, some retained 𝐽 satisfies Δ(𝐻∗ , 𝐽) ≤ 𝑇 ( 𝐴), and Proposition 4.5 gives 𝜈 𝐽 (𝜋∗ ) ≤ ℎ+ ( 𝐴). Equation (4) gives the squeezing bound. Every retained basis has true cutoff mass at least 𝑤 = 7𝛼/16, and distinct retained bases have projector 34

overlap at most 𝜇 = 𝛼/16. Lemma 4.6 bounds their number by 1−𝜇 8 ≤ . 𝑤 − 𝜇 3𝛼 Thus the failure condition is never triggered on the stated success event.

□

The full algorithm. The full algorithm runs 𝑅 blocks, starting from the list that contains only the Fock basis 𝐺 = 𝐼, and updates the squeezing bound by 𝐴 ↦→ 𝐴+ ( 𝐴) after each block. Given 𝛼, 𝐴0 , 𝜂 as in Theorem 4.1, set   𝑅 = log2 (𝑒 + log 𝐴0 ) ,

𝑁 𝛼 = 13𝑊 𝐿 12 𝛼,

(48)

and choose 𝑏 = ⌈2 log(2𝑁 𝛼 𝑅/𝜂) + 800⌉ ,

𝐽 = 𝑁 𝛼 𝑅𝑏 12 .

Set 𝜏 = 𝜂/(2𝐽). Here 𝑅 is the number of blocks, 𝑏 is the number of decoder runs in each union of lists, and 𝐽 bounds the total number of unions of lists and cutoff-mass estimates over the whole run, as shown in the proof below. Algorithm 2 applies 𝑅 blocks with these choices of 𝑏 and 𝜏. Algorithm 2: List-decodable iterative unsqueezing Input: Copies of 𝜌, 𝛼 ∈ (0, 1], an initial bound 𝐴0 ≥ 1, and 𝜂 ∈ (0, 1/2]. Output: A list of at most 𝑊 bases, or failure. 1 Set L0 = {𝐼}; 2 for 𝑟 = 0, . . . , 𝑅 − 1 do 3 Run Algorithm 1 with (𝜌, 𝛼, L𝑟 , 𝐴𝑟 , 𝑏, 𝜏) to obtain L𝑟+1 ; 4 if the block reports failure then 5 Return failure; 6

Set 𝐴𝑟+1 = 𝐴+ ( 𝐴𝑟 );

7 Return L 𝑅 ;

Proof of Theorem 4.1. We analyze Algorithm 2 in four steps: we count the random subroutine calls, show that with probability at least 1 − 𝜂 all of them succeed, follow a useful basis through the blocks on that event, and bound the cost. Counting calls. Each union of 𝑏 lists has at most 𝐿 𝛼 𝑏 pairs, so every basis has at most 𝐿 𝛼 𝑏 updated bases. Starting from at most 𝑊 bases, round 𝑗 of a block holds at most 𝑊 (𝐿 𝛼 𝑏) 𝑗 bases. A block computes one union of lists for each basis in rounds 0, . . . , 𝑞, at most (𝑞 + 1)𝑊 (𝐿 𝛼 𝑏) 𝑞 = 12𝑊 (𝐿 𝛼 𝑏) 11 in total, and one cutoff-mass estimate for each of the at most 𝑊 (𝐿 𝛼 𝑏) 12 candidates. Over 𝑅 blocks, the total number of unions and mass estimates is therefore at most 13𝑅𝑊 (𝐿 𝛼 𝑏) 12 = 𝑁 𝛼 𝑅𝑏 12 = 𝐽, including on runs that stop early with failure. All calls succeed. By Lemma 4.2, each union of 𝑏 lists fails to contain a good pair with probability at most 10−𝑏 ≤ 𝑒 −𝑏 , and each mass estimate fails with probability at most 𝜏 = 𝜂/(2𝐽). Both bounds hold conditionally on all earlier observations, because every call uses fresh copies, so a union bound over the at

35

most 𝐽 calls applies. Our choice of 𝑏 gives 12 log 𝑏 ≤ 𝑏/2, and therefore 𝐽𝑒 −𝑏 = 𝑁 𝛼 𝑅𝑏 12 𝑒 −𝑏 ≤ 𝑁 𝛼 𝑅𝑒 −𝑏/2 ≤ 𝜂/2. So with probability at least 1−𝜂, every union of lists contains a good pair and every mass estimate is accurate. Following a useful basis. Condition on this event. The Fock basis 𝐺 = 𝐼 is useful at level 𝐴0 . By Lemma 4.8 and induction, every list L𝑟 contains a basis useful at level 𝐴𝑟 , and the algorithm never reports failure. It remains to bound 𝐴 𝑅 . Write 𝑝 = 8 and 𝑑 = 32 for the exponents in Lemma 4.3 and Proposition 4.5, −𝑞 −𝑞 and set 𝑣 𝛼 = 2 𝑝 𝑢 𝛼 Λ2 𝑝 ≍ 𝑡 2𝛼𝑝+6 . Substituting 𝐵( 𝐴) = 2Λ2 𝐴2 into ℎ 𝛼 gives ℎ( 𝐴) = 𝑣 𝛼 𝐴 𝑝2 . Since 𝑘 ( 𝐴) ≲ ℎ( 𝐴)/𝛼 and 𝜇 ≍ 𝛼, we have 𝑇 ( 𝐴) ≲ 𝛼 −2𝑑 ℎ( 𝐴) 𝑑 . Since 𝐴+ ( 𝐴) = 48𝐶tr𝑇 ( 𝐴) 3 ℎ( 𝐴), we obtain, for a sufficiently large universal 𝐶, K 𝛼 = 𝐶𝛼 −6𝑑 𝑣 3𝑑+1 ≍ 𝛼 −𝑠/2 , 𝛼

and 𝐴+ ( 𝐴) ≤ K 𝛼 𝐴 𝑝 (3𝑑+1)2

−𝑞

𝑠 ≍ 𝑟 dec ,

(49)

√ ≤ K 𝛼 𝐴.

Here 𝑠 ≍ 𝑟 dec because 𝑝 and 𝑑 are universal and 𝑡 𝛼 ≍ 𝛼 −𝑟dec . The last inequality uses 𝑝(3𝑑 + 1)2−𝑞 = 776/2048 ≤ 1/2; this is why we take 𝑞 = 11, the smallest value for which it holds. Taking logarithms, each block maps log 𝐴 to at most 21 log 𝐴 + log K 𝛼 , so log 𝐴𝑟 ≤ 2−𝑟 log 𝐴0 + 2 log K 𝛼 . By (48), 2−𝑅 log 𝐴0 ≤ 1, so 𝐴 𝑅 ≤ 𝐻 𝛼 with 𝐻 𝛼 = 𝑒 K2𝛼 ≍ 𝛼 −𝑠 ,

𝑠 ≍ 𝑟 dec .

(50)

Since 𝑅 ≥ 1, the last block also gives a kept basis 𝐺 with 𝜈𝐺 (𝜋∗ ) ≤ ℎ+ ( 𝐴 𝑅−1 ) ≤ 𝐴 𝑅 ≤ 𝐻 𝛼 . This proves (36) and the success probability. Cost. Each union of lists uses at most 𝑏⌈𝑒 𝑡 𝛼 (3𝑛) 𝑡 𝛼 ⌉ copies and 𝑏 decoder calls. Each mass estimate uses 𝑂 (𝛼 −2 log(𝐽/𝜂)) copies, independently of its photon cutoff. The remaining computations, namely Williamson decompositions, projections onto K 𝐴, and evaluations of Δ by Lemma 4.4, take polynomially many arithmetic operations in 𝑛 per candidate, and each candidate is compared with at most 𝑊 kept bases. Enlarging 𝐶dec if necessary, we may assume that 𝑡 𝛼 exceeds the degree of these polynomials. Since 𝐿 𝛼 = ⌈𝑒 𝑡 𝛼 ⌉, we have 𝑁 𝛼 ≤ exp(𝐶𝑡 𝛼 ) and 𝑏 ≤ 𝐶𝑡 𝛼 log(𝑅/𝜂). Multiplying these costs by the call bound 𝐽 = 𝑁 𝛼 𝑅𝑏 12 shows that the number of copies, the number of decoder calls, and the number of arithmetic operations outside the decoder are each at most exp(𝐶𝑡 𝛼 )𝑛𝑡 𝛼 𝑅 log(𝑅/𝜂)

13

.

Since 𝑅 ≲ log log(𝑒 + 𝐴0 ), this proves (37). □

36

4.2

Optimization at bounded photon number

The preceding subsection supplies a list containing a basis in which a best Gaussian approximator has total mean photon number at most ℎ. We now construct an optimizer that competes with every pure Gaussian state satisfying this photon bound. The input 𝜎 is arbitrary. Theorem 4.9. Let 𝜎 be an 𝑛-mode state, ℎ ≥ 1, 𝜖 ∈ (0, 1/2], and 𝜂 ∈ (0, 1/2]. There is an algorithm which outputs a pure Gaussian state 𝜋ˆ satisfying 𝐹 (𝜎, 𝜋) ˆ ≥

sup

𝐹 (𝜎, 𝜋) − 𝜖

𝜋 ∈ G:𝜈 ( 𝜋 ) ≤ℎ

with probability at least 1 − 𝜂, using at most 3

𝐶ℎ, 𝜖 𝑛𝑂 (ℎ +ℎ log(1/𝜖 ) ) log(2/𝜂)

(51)

copies and runtime. Here 𝐶ℎ, 𝜖 is a constant depends only on ℎ and 𝜖. Moreover, the output has mean photon number 𝜈( 𝜋) ˆ at most 17(ℎ + 1) 3 . 4.2.1

Setup

Write 𝜋 𝑏, 𝐴 = |𝜓(𝑏, 𝐴)⟩⟨𝜓(𝑏, 𝐴)| for the pure Gaussian state in (5). We optimize over the following convex set of Bargmann parameters: √︂ n √ o √ ℎ 𝑟ℎ = (52) , Dℎ = (𝑏, 𝐴) ∈ P𝑛 : ∥ 𝐴∥ op ≤ 𝑟 ℎ , ∥ 𝐴∥ 𝐹 ≤ ℎ, ∥𝑏∥ 2 ≤ 2 ℎ . ℎ+1 Here ℎ ≥ 1 and 𝐴 = 𝐴⊤ by the definition of P𝑛 . Since 𝑟 ℎ < 1, every point of Dℎ defines a normalizable pure Gaussian state. We will show that this domain contains all pure Gaussian states with total photon number at most ℎ, while its other points still have mean photon number bounded independently of 𝑛. Lemma 4.10 (A convex domain for best Gaussian approximators with bounded photon number). For ℎ ≥ 1, the set Dℎ in (52) is convex and consists of valid pure Gaussian parameters. Every pure Gaussian state 𝜋 𝑏, 𝐴 with 𝜈(𝜋 𝑏, 𝐴) ≤ ℎ has (𝑏, 𝐴) ∈ Dℎ . Conversely, every (𝑏, 𝐴) ∈ Dℎ satisfies 𝜈(𝜋 𝑏, 𝐴) ≤ ℎ(ℎ + 1) + 16ℎ(ℎ + 1) 2 ≤ 17(ℎ + 1) 3 . We defer the proof to Appendix A.3.3. We next study the normalization 𝑍 (𝑏, 𝐴) = ∥Φ(𝑏, 𝐴) ∥ 2 from (5). Lemma 2.6 lets us compute 𝑍 and its derivatives using real matrices; we now use it to bound log 𝑍 and its gradient on Dℎ . Lemma 4.11 (Convexity and bounds for the log normalization). The function 𝐿 (𝑏, 𝐴) = log 𝑍 (𝑏, 𝐴) is smooth and convex on the real convex domain {(𝑏, 𝐴) ∈ P𝑛 : ∥ 𝐴∥ op < 1}. For every (𝑏, 𝐴) ∈ Dℎ , 0 ≤ 𝐿 (𝑏, 𝐴) ≤ 𝐵 ℎ ≡ 17(ℎ + 1) 2 , √ 6 ℎ 8ℎ ∥∇𝐿 (𝑏, 𝐴) ∥ P ≤ 𝐺 ℎ ≡ + . 1 − 𝑟 ℎ (1 − 𝑟 ℎ ) 2

37

Here 𝐵 ℎ ≍ (ℎ + 1) 2 and 𝐺 ℎ ≍ (ℎ + 1) 3 for ℎ ≥ 1. The gradient uses the real inner product on P𝑛 , whose norm is ∥(𝑣 𝑏 , 𝑣 𝐴) ∥ 2P = ∥𝑣 𝑏 ∥ 22 + 12 ∥𝑣 𝐴 ∥ 2𝐹 . 2

Proof. Convexity. Think of (10) as integration against the measure 𝑑𝜇(𝑧) = 𝜋 −𝑛 𝑒 − ∥ 𝑧 ∥ 2 𝑑 2𝑛 𝑧 and write the exponent in the integral as as ℓ 𝜃 (𝑧) = 2 Re(𝑏 ⊤ 𝑧) + Re(𝑧⊤ 𝐴𝑧), where 𝜃 = (𝑏, 𝐴). This is real-linear in 𝜃. For 0 < 𝑡 < 1, Hölder’s inequality with exponents 1/(1 − 𝑡) and 1/𝑡 gives ∫ ′ 𝑍 ((1 − 𝑡)𝜃 + 𝑡𝜃 ) = (𝑒 ℓ 𝜃 ) 1−𝑡 (𝑒 ℓ 𝜃 ′ ) 𝑡 𝑑𝜇 ≤ 𝑍 (𝜃) 1−𝑡 𝑍 (𝜃 ′ ) 𝑡 . Convexity follows immediately. Smoothness follows from (11), since 𝑀 ( 𝐴) is positive definite when ∥ 𝐴∥ op < 1. Value bound. Let 𝑟 = ∥ 𝐴∥ op < 1. Lemma 2.6 implies that 𝐿 (𝑏, 𝐴) = −

  1 ∑︁ log 1 − 𝑠2𝑗 + 𝑢 ⊤ 𝑀 −1 𝑢. 2 𝑗

Both terms are nonnegative. Using − log(1 − 𝑥) ≤ 𝑥/(1 − 𝑥) for 0 ≤ 𝑥 < 1, ∥𝑢∥ 2 = ∥𝑏∥ 2 , and ∥ 𝑀 −1 ∥ op = (1 − 𝑟) −1 , we obtain ∥𝑏∥ 22 ∥ 𝐴∥ 2𝐹 0 ≤ 𝐿 (𝑏, 𝐴) ≤ + . (53) 2(1 − 𝑟 2 ) 1 − 𝑟 On Dℎ , recall that 1/(1 − 𝑟 ℎ2 ) = ℎ + 1 and 1/(1 − 𝑟 ℎ ) = (1 + 𝑟 ℎ ) (ℎ + 1) ≤ 2(ℎ + 1) to bound the RHS by 1 ℎ(ℎ + 1) + 8ℎ(ℎ + 1) ≤ 17(ℎ + 1) 2 . 2 Gradient bound. We bound the directional derivative in an arbitrary real direction 𝑣 = (𝑣 𝑏 , 𝑣 𝐴) with ∥𝑣∥ P = 1. Set     − Re 𝑣 𝐴 Im 𝑣 𝐴 Re 𝑣 𝑏 𝐻 = 𝐷 𝑀 ( 𝐴) [𝑣 𝐴] = , 𝑞 = 𝐷𝑢(𝑏) [𝑣 𝑏 ] = . Im 𝑣 𝐴 Re 𝑣 𝐴 − Im 𝑣 𝑏 The block entries and the definition of the parameter norm imply √ tr 𝐻 = 0, ∥𝐻 ∥ 𝐹 = 2∥𝑣 𝐴 ∥ 𝐹 ≤ 2, ∥𝑞∥ 2 = ∥𝑣 𝑏 ∥ 2 ≤ 1. Differentiating (11), using 𝐷 (𝑀 −1 ) [𝐻] = −𝑀 −1 𝐻 𝑀 −1 , gives   𝐷 𝐿 (𝑏, 𝐴) [𝑣] = − 12 tr 𝑀 −1 𝐻 + 2𝑞 ⊤ 𝑀 −1 𝑢 − 𝑢 ⊤ 𝑀 −1 𝐻 𝑀 −1 𝑢.

(54)

√ Write 𝛿 = 1−𝑟 ℎ > 0. On Dℎ , ∥ 𝑀 −1 ∥ op ≤ 𝛿 −1 and ∥𝑢∥ 2 ≤ 2 ℎ. To bound the trace term without introducing a factor depending on 𝑛, use tr 𝐻 = 0 to replace 𝑀 −1 by 𝑀 −1 − 𝐼. Since the block formula for 𝑀 gives √ ∥𝐼 − 𝑀 ∥ 𝐹 = 2∥ 𝐴∥ 𝐹 , √ 2ℎ −1 −1 ∥ 𝑀 − 𝐼 ∥ 𝐹 = ∥ 𝑀 (𝐼 − 𝑀) ∥ 𝐹 ≤ . 𝛿

38

The Frobenius Cauchy–Schwarz inequality now yields 

−1 1 2 | tr 𝑀 𝐻





| = 21 | tr (𝑀 −1 − 𝐼)𝐻



| ≤ 21 ∥ 𝑀 −1 − 𝐼 ∥ 𝐹 ∥𝐻 ∥ 𝐹 ≤

√ 2ℎ . 𝛿

The other two terms in (54) satisfy √ 4 ℎ , |2𝑞 𝑀 𝑢| ≤ 2∥𝑞∥ 2 ∥ 𝑀 ∥ op ∥𝑢∥ 2 ≤ 𝛿 8ℎ |𝑢 ⊤ 𝑀 −1 𝐻 𝑀 −1 𝑢| ≤ ∥𝐻 ∥ op ∥ 𝑀 −1 𝑢∥ 22 ≤ 2 , 𝛿 √ where ∥𝐻 ∥ op ≤ ∥𝐻 ∥ 𝐹 ≤ 2. Adding these bounds and using 2 + 4 ≤ 6 gives |𝐷 𝐿 (𝑏, 𝐴) [𝑣] | ≤ 𝐺 ℎ . Finally, ∥∇𝐿(𝑏, 𝐴)∥ P = sup ∥ 𝑣 ∥ P =1 |𝐷 𝐿 (𝑏, 𝐴) [𝑣] |, proving the gradient bound. □ ⊤

4.2.2

−1

−1

Learning the low degree components

Let 𝑃𝐾 = 1 { 𝑁 ≤𝐾 } = Π ≤𝐾 project onto the Fock basis vectors with total photon number at most 𝐾. In Bargmann space, this keeps precisely the monomials of total degree at most 𝐾. We first choose a cutoff that approximates every best Gaussian approximator in Dℎ , then learn the finite block 𝜎𝐾 = 𝑃𝐾 𝜎𝑃𝐾 of the input state. The cutoff applies to the best Gaussian approximators; the input 𝜎 can have arbitrary photon number. Lemma 4.12 (Uniform photon cutoff). Let ℎ ≥ 1 and 0 < 𝜖 ≤ 1/2. With 𝐵 ℎ = 17(ℎ + 1) 2 as in Lemma 4.11, set   1 𝐵 ℎ + 2 log(64/𝜖) 𝑠 ℎ = log(1 + 1/ℎ), 𝐾 = 𝐾 ℎ, 𝜖 = . (55) 4 𝑠ℎ Thus 𝑠 ℎ ≍ (ℎ + 1) −1 and 𝐾 = 𝐾 ℎ, 𝜖 ≍ (ℎ + 1) 3 + (ℎ + 1) log(1/𝜖). For every 𝜃 = (𝑏, 𝐴) ∈ Dℎ , writing |𝜓 𝜃 ⟩ = |𝜓(𝑏, 𝐴)⟩ and 𝜋 𝜃 = |𝜓 𝜃 ⟩⟨𝜓 𝜃 |, we have ∥ (𝐼 − 𝑃𝐾 ) |𝜓 𝜃 ⟩ ∥ 2 ≤ (𝜖/64) 2 . Consequently, for every density operator 𝜎, |𝐹 (𝜎, 𝜋 𝜃 ) − ⟨𝜓 𝜃 | 𝑃𝐾 𝜎𝑃𝐾 |𝜓 𝜃 ⟩| ≤ 𝜖/16.

(56)

Proof. Our goal will be to prove ⟨𝜓 𝜃 | 𝑒 𝑠ℎ 𝑁 |𝜓 𝜃 ⟩ ≤ 𝑒 𝐵ℎ

(𝜃 ∈ Dℎ ).

(57)

We first deduce the lemma from this inequality, then prove the inequality. Proof of lemma using (57) Since 𝑁 has nonnegative integer eigenvalues, 𝐼 − 𝑃𝐾 ⪯ 𝑒 −𝑠ℎ (𝐾+1) 𝑒 𝑠ℎ 𝑁 . Thus 𝑡 𝜃 ≡ ∥ (𝐼 − 𝑃𝐾 ) |𝜓 𝜃 ⟩ ∥ 2 ≤ 𝑒 −𝑠ℎ (𝐾+1) ⟨𝜓 𝜃 | 𝑒 𝑠ℎ 𝑁 |𝜓 𝜃 ⟩ ≤ 𝑒 𝐵ℎ −𝑠ℎ (𝐾+1) ≤ (𝜖/64) 2 . To prove (56), let |𝜙⟩ = 𝑃𝐾 |𝜓 𝜃 ⟩ and | 𝜒⟩ = (𝐼 − 𝑃𝐾 ) |𝜓 𝜃 ⟩. Then ⟨𝜓 𝜃 | 𝜎 |𝜓 𝜃 ⟩ − ⟨𝜙| 𝜎 |𝜙⟩ = ⟨𝜒| 𝜎 |𝜓 𝜃 ⟩ + ⟨𝜙| 𝜎 | 𝜒⟩ . 39

The magnitude of this is at most 𝜖/16. Proof of (57) Multiplying each 𝑚-photon coefficient by 𝑒 𝑠𝑚/2 corresponds in Bargmann space to replacing ⊤ ⊤ 𝑧 by 𝑒 𝑠/2 𝑧. Applied to 𝑒 𝑏 𝑧+𝑧 𝐴𝑧/2 , this replaces (𝑏, 𝐴) by (𝑒 𝑠/2 𝑏, 𝑒 𝑠 𝐴). Taking squared norms therefore gives 𝑍 (𝑒 𝑠/2 𝑏, 𝑒 𝑠 𝐴) ⟨𝜓(𝑏, 𝐴)| 𝑒 𝑠 𝑁 |𝜓(𝑏, 𝐴)⟩ = , 𝑒 𝑠 ∥ 𝐴∥ op < 1. 𝑍 (𝑏, 𝐴) The condition ensures that the scaled Bargmann function is square integrable, so this also justifies finiteness of the expectation. √︁ Write 𝑟 = 𝑟 ℎ = ℎ/(ℎ + 1) and take 𝑠 = 𝑠 ℎ = − 12 log 𝑟. For 𝑏˜ = 𝑒 𝑠/2 𝑏 and 𝐴˜ = 𝑒 𝑠 𝐴, the domain constraints give √ ˜ op ≤ 𝑟 < 1, ∥ 𝐴∥

√ ˜ 2 ≤ 4ℎ/ 𝑟 ≤ 4(ℎ + 1). ∥ 𝑏∥ 2

˜ 2 ≤ ℎ/𝑟 ≤ ℎ + 1, ∥ 𝐴∥ 𝐹

√ √ Also (1 − 𝑟) −1 = (1 + 𝑟) (ℎ + 1) ≤ 2(ℎ + 1) and (1 − 𝑟) −1 = (1 + 𝑟)/(1 − 𝑟) ≤ 4(ℎ + 1). The general normalization bound (53), which does not require the scaled parameters to lie in Dℎ , now yields ˜ 𝐴) ˜ ≤ log 𝑍 ( 𝑏,

ℎ+1 4(ℎ + 1) + √ ≤ 17(ℎ + 1) 2 = 𝐵 ℎ . 2(1 − 𝑟) 1− 𝑟

Since 𝑍 (𝑏, 𝐴) ≥ 1, we are done.

□

The cutoff space has dimension 𝐷 𝐾 = #{k ∈ N𝑛 :

Í

𝑗 𝑘 𝑗 ≤ 𝐾} =

𝑛+𝐾
≤ (𝑛 + 1) 𝐾 . 𝐾

We learn 𝜎𝐾 as an unnormalized positive matrix of trace at most one. Lemma 4.13 (Tomography of the cutoff block). Let 𝐾 ≥ 0 be an integer, 0 < 𝜖 ≤ 1/2, and 0 < 𝜂 ≤ 1/2. Given copies of an arbitrary state 𝜎, measurements on individual copies suffice to construct a matrix 𝜎 ˆ𝐾 supported on the cutoff space such that, with probability at least 1 − 𝜂, 𝜎 ˆ 𝐾 ⪰ 0,

tr 𝜎 ˆ 𝐾 ≤ 1,

∥𝜎 ˆ 𝐾 − 𝑃𝐾 𝜎𝑃𝐾 ∥ op ≤ 𝜖/16.

The number of copies is 𝑂 (𝐷 4𝐾 𝜖 −2 log(2𝐷 𝐾 /𝜂)). We defer the proof to Appendix A.3.4. Together these lemmas give a uniformly accurate objective on Dℎ . Extend 𝜎 ˆ 𝐾 by zero outside the cutoff space and define 𝑝(𝜃) 𝑝(𝜃) = ⟨Φ(𝜃)| 𝜎 ˆ 𝐾 |Φ(𝜃)⟩ , 𝑓 (𝜃) = = ⟨𝜓 𝜃 | 𝜎 ˆ 𝐾 |𝜓 𝜃 ⟩ . (58) 𝑍 (𝜃) On the success event, the triangle inequality and ∥𝑃𝐾 |𝜓 𝜃 ⟩ ∥ ≤ 1 imply | 𝑓 (𝜃) − 𝐹 (𝜎, 𝜋 𝜃 )| ≤ 𝜖/8

for every 𝜃 ∈ Dℎ .

(59)

Also 0 ≤ 𝑓 ≤ 1 because 𝜎 ˆ 𝐾 ⪰ 0 and tr 𝜎 ˆ 𝐾 ≤ 1. Each Fock coefficient of 𝑃𝐾 |Φ(𝜃)⟩ is a Laurent polynomial in (𝑏, 𝐴) of degree at most 𝐾, by expanding the exponential in (5). Thus 𝑝 is a real polynomial of degree at

40

most 2𝐾 in the real and imaginary parts of the parameters. This is the finite polynomial numerator we will optimize, with log 𝑍 controlling the denominator. 4.2.3

Global optimization of the normalized overlap

It remains to maximize 𝑓 = 𝑝/𝑍 from (58) over Dℎ . Write ℓ = log 𝑍. By Lemma 4.11, ℓ is convex, 0 ≤ ℓ ≤ 𝐵 ℎ , and ∥∇ℓ∥ P ≤ 𝐺 ℎ on this domain. Unfortunately, 𝑝 does not satisfy any nice concavity properties, and we are not aware of any local optimization methods that are useful for global optimization. Instead, we will find a subspace of bounded dimension that determines 𝑝 up to small error. We will then brute force over a fine-enough net of this subspace and use standard tools from convex optimization to minimize the 𝑍 in the orthogonal subspace. Theorem 4.14 (Optimization of the normalized overlap). Fix ℎ ≥ 1, an integer 𝐾 ≥ 1, and 0 < 𝛿 ≤ 1/2. Given a matrix 𝜎 ˆ 𝐾 ⪰ 0 supported on 𝑃𝐾 F𝑛 with tr 𝜎 ˆ 𝐾 ≤ 1, there exists a deterministic algorithm that returns 𝜃ˆ ∈ Dℎ such that ˆ ≥ max 𝑓 (𝜃) − 𝛿. 𝑓 ( 𝜃) 𝜃 ∈ Dℎ

It uses at most 𝐶ℎ,𝐾 , 𝛿 (𝑛 + 1) 𝑐𝐾 real-arithmetic operations, for a universal constant 𝑐. One sufficient choice of the prefactor is Γ = (1 + 5ℎ) 𝐾 (𝐾 + 1) 4 𝐾!,    2 𝐶 (ℎ + 1) (𝐾 + 1)Γ 6Γ 𝐶ℎ,𝐾 , 𝛿 = exp 𝐶 (𝐾 + 1) 2 log , 𝛿 𝛿

(60)

for a sufficiently large universal 𝐶. These are sufficient, nonoptimized choices for the constants in the stated algorithm. Reducing the numerator to a tensor polynomial. The next theorem gives the precise reduction we need. Its proof, including the coefficient-norm bound, is deferred to Appendix B. Theorem 4.15 (Tensor representation and effective coordinates). Let 𝐻 = C𝑛 and let Sym2 𝐻 be the symmetric subspace of 𝐻 ⊗2 with its inherited norm. Identify a symmetric matrix with its coefficient tensor, so its tensor norm is its Frobenius norm. Put √ X = C ⊕ 𝐻 ⊕ Sym2 𝐻, 𝑅 ℎ = 1 + 5ℎ, √ (1, 𝑏, 𝐴/ 2) 4 𝑦(𝑏, 𝐴) = , Γ = 𝑅 2𝐾 ℎ (𝐾 + 1) 𝐾!. 𝑅ℎ From 𝜎 ˆ 𝐾 one can construct a positive semidefinite operator 𝐵 on X ⊗𝐾 with ∥𝐵∥ 𝐹 ≤ Γ such that 𝑝(𝜃) = 𝑔(𝑦(𝜃)),

𝑔(𝑣) = ⟨𝑣 ⊗𝐾 , 𝐵𝑣 ⊗𝐾 ⟩.

For 𝜃, 𝜃 ′ ∈ Dℎ , ∥𝑦(𝜃) ∥ 2 ≤ 1,

∥𝑦(𝜃) − 𝑦(𝜃 ′ ) ∥ 2 =

41

∥𝜃 − 𝜃 ′ ∥ P . 𝑅ℎ

(61)

The polynomial 𝑔 is 2𝐾Γ-Lipschitz on the unit ball of X. For every 0 < 𝜉 < Γ, one can also construct a complex subspace 𝑊 ⊆ X satisfying dimC 𝑊 ≤ 8(𝐾 + 1) 6 (Γ/𝜉) 2 ,

sup |𝑔(𝑣) − 𝑔(𝑃𝑊 𝑣)| ≤ 𝜉.

(62)

∥ 𝑣 ∥ 2 ≤1

The constructions cost at most 𝐶 (𝐾 + 1) 𝐶 𝐾!(𝑛 + 1) 𝑐𝐾 arithmetic operations for universal 𝐶, 𝑐, counting spectral factorizations in the real-arithmetic model. The accuracy 𝜉 sets the singular-value threshold and the dimension bound in (62); it introduces no further factor in this construction cost. Thus 𝑝 is a tensor quadratic form evaluated on the image 𝑦(Dℎ ). The approximation in (62) gives a subspace with dimension independent of 𝑛 such that the action of 𝐵 is essentially determined by its action on that subspace. Choosing the denominator by convex optimization. Let 𝑊 and 𝜉 be the parameters from Theorem 4.15, and let Λ = max{1, 2𝐾Γ}. The following lemma formalizes the idea that to approximately maximize 𝑓 = 𝑝/𝑍, it suffices to brute force over a net of the subspace 𝑊, while doing convex optimization to minimize the denominator at each point of the net. Let [𝑥] + = max{0, 𝑥}. Lemma 4.16 (Search over numerator coordinates). Set 𝑡=

𝜉 , 3Λ

𝜆=

𝐵ℎ + 1 , 𝑡

𝐽𝑧 (𝜃) = ℓ(𝜃) + 𝜆[∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 − 𝑡] + .

(63)

Let N be a finite 𝑡-net of the unit ball of 𝑊. For each 𝑧 ∈ N , let 𝜃 𝑧 ∈ Dℎ be such that 𝐽𝑧 (𝜃 𝑧 ) ≤ min 𝐽𝑧 (𝜃) + 𝜁, 𝜃 ∈ Dℎ

0 < 𝜁 ≤ 1.

Let 𝑧ˆ ∈ argmax 𝑓 (𝜃 𝑧 ),

𝜃ˆ = 𝜃 𝑧ˆ .

𝑧∈N

Then ˆ ≥ max 𝑓 (𝜃) − (3𝜉 + 𝜁). 𝑓 ( 𝜃)

(64)

𝜃 ∈ Dℎ

Moreover, each 𝐽𝑧 is convex on Dℎ . Proof. The domain Dℎ is compact and 𝑓 is continuous, so it has a maximizer 𝜃 ∗ . Choose 𝑧 ∈ N with ∥𝑃𝑊 𝑦(𝜃 ∗ ) − 𝑧∥ 2 ≤ 𝑡. This makes the penalty at 𝜃 ∗ zero, and hence 𝐽𝑧 (𝜃 𝑧 ) ≤ 𝐽𝑧 (𝜃 ∗ ) + 𝜁 = ℓ(𝜃 ∗ ) + 𝜁 ≤ 𝐵 ℎ + 1. Both terms of 𝐽𝑧 are nonnegative. Bounding each of them inidividually by 𝐵 ℎ + 1 yields ℓ(𝜃 𝑧 ) ≤ ℓ(𝜃 ∗ ) + 𝜁,

∥𝑃𝑊 𝑦(𝜃 𝑧 ) − 𝑧∥ 2 ≤ 𝑡 +

𝐵ℎ + 1 = 2𝑡. 𝜆

(65)

The projected coordinates of 𝜃 𝑧 and 𝜃 ∗ are therefore at distance at most 3𝑡. Applying (62) at both points,

42

followed by the Lipschitz bound for 𝑔, yields | 𝑝(𝜃 𝑧 ) − 𝑝(𝜃 ∗ )| ≤ 2𝜉 + 3Λ𝑡 = 3𝜉. Since 𝑝 ≥ 0, the denominator bound in (65) implies ∗

𝑓 (𝜃 𝑧 ) ≥ 𝑒 −ℓ ( 𝜃 ) −𝜁 𝑝(𝜃 𝑧 ) ∗

≥ 𝑒 −𝜁 𝑓 (𝜃 ∗ ) − 3𝜉𝑒 −ℓ ( 𝜃 ) −𝜁 ≥ 𝑓 (𝜃 ∗ ) − 𝜁 − 3𝜉. ˆ ≥ 𝑓 (𝜃 𝑧 ), proving (64). Here we used 0 ≤ 𝑓 ≤ 1, ℓ ≥ 0, and 1 − 𝑒 −𝜁 ≤ 𝜁. By the choice of 𝑧ˆ, 𝑓 ( 𝜃) 𝐽𝑧 is a sum of convex functions, so it is convex. Convex minimization.

□

We use Theorem 3.2 of Bubeck [Bub15] as a black box in the following form.

Fact 4.17 (Projected subgradient descent).√︁Let H be a finite-dimensional real Hilbert space with inner product ⟨·, ·⟩ H and induced norm ∥𝑥∥ H = ⟨𝑥, 𝑥⟩ H . Let 𝐷 ⊆ H be compact and convex, with 0 ∈ 𝐷 and ∥𝜃 ∥ H ≤ 𝐷 0 on 𝐷, where 𝐷 0 > 0. Let 𝐽 : 𝐷 → R be continuous and convex. Suppose an oracle returns, at every 𝜃 ∈ 𝐷, a subgradient 𝑣 ∈ 𝜕𝐽 (𝜃) with ∥𝑣∥ H ≤ 𝐺, where 𝐺 > 0; that is, 𝐽 (𝜃 ′ ) ≥ 𝐽 (𝜃) + ⟨𝑣, 𝜃 ′ − 𝜃⟩ H

for every 𝜃 ′ ∈ 𝐷.

Suppose we also have an oracle for orthogonal projection onto 𝐷 in this Hilbert-space norm: proj𝐷 (𝑥) = argmin ∥𝑥 − 𝑦∥ 2H . 𝑦 ∈𝐷

Given an accuracy 𝜁 > 0, let 𝑇 = max{1, ⌈(𝐷 0 𝐺/𝜁) 2 ⌉}, 𝛼 = 𝐷√0 . Now consider the following algorithm. 𝐺 𝑇 Initialize 𝜃 0 = 0, and consider the iterates 𝜃 𝑗+1 = proj𝐷 (𝜃 𝑗 − 𝛼𝑣 𝑗 ), where 𝑣 𝑗 is returned by the subgradient Í −1 oracle. Then 𝜃¯ = 𝑇 −1 𝑇𝑗=0 𝜃 𝑗 ∈ 𝐷 satisfies ¯ ≤ min 𝐽 + 𝐷√0 𝐺 ≤ min 𝐽 + 𝜁 . 𝐽 ( 𝜃) 𝐷 𝐷 𝑇 Clearly, this algorithm requires 𝑇 subgradient and 𝑇 projection oracle queries. We apply this convex optimization algorithm to the functions 𝐽𝑧 (𝜃) using the efficient subgradient and projection oracles in the following proposition. Proposition 4.18 (Oracles for the Gaussian optimization problem). For every net point 𝑧, the function 𝐽𝑧 in (63) has a subgradient oracle on Dℎ with norm bound 𝐺 ℎ + 𝜆/𝑅 ℎ . Projection onto Dℎ in the parameter norm is also computable. Both operations have polynomial arithmetic cost in 𝑛 and dimC 𝑊. The proof and efficient oracle implementations are given in Appendix C. We now have all the pieces in place to describe the full algorithm; see the description of Algorithm 3. It takes in the estimate of 𝑃𝐾 𝜎𝑃𝐾 , and outputs the parameters of a pure Gaussian state that approximately maximizes fidelity with 𝜎.

43

Algorithm 3: Global optimization of the normalized Gaussian overlap Input: ℎ ≥ 1, 𝐾 ≥ 1, 0 < 𝛿 ≤ 1/2, and 𝜎 ˆ 𝐾 ⪰ 0 with tr 𝜎 ˆ 𝐾 ≤ 1 supported on 𝑃𝐾 F𝑛 . ˆ ≥ max Dℎ 𝑓 − 𝛿. Output: 𝜃ˆ ∈ Dℎ with 𝑓 ( 𝜃) 1 Set 𝜉 = 𝛿/6 and 𝜁 = 𝛿/2; 2 Construct 𝐵, 𝑦, Γ, and the subspace 𝑊 at accuracy 𝜉 using Theorem 4.15; set 𝑔(𝑣) = ⟨𝑣 ⊗𝐾 , 𝐵𝑣 ⊗𝐾 ⟩; 3 Set Λ = max{1, 2𝐾Γ}, 𝑡 = 𝜉/(3Λ), and 𝜆 = (𝐵 ℎ + 1)/𝑡; 4 Construct a 𝑡-net N of the unit ball of 𝑊; √ √ 5 Set 𝐷 0 = 5ℎ, 𝐺 = 𝐺 ℎ + 𝜆/𝑅 ℎ , 𝑇 = max{1, ⌈(𝐷 0 𝐺/𝜁) 2 ⌉}, and 𝛼 = 𝐷 0 /(𝐺 𝑇); 6 foreach 𝑧 ∈ N do 7 Define 𝐽𝑧 (𝜃) = ℓ(𝜃) + 𝜆[∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 − 𝑡] + and initialize 𝜃 0 = 0; 8 for 𝑗 = 0, . . . , 𝑇 − 1 do 9 Compute 𝑣 𝑗 ∈ 𝜕𝐽𝑧 (𝜃 𝑗 ) with ∥𝑣 𝑗 ∥ P ≤ 𝐺 using Proposition 4.18; 10 Set 𝜃 𝑗+1 = proj Dℎ (𝜃 𝑗 − 𝛼𝑣 𝑗 ) using (105) and radial projection of 𝑏; Í −1 𝜃 𝑗; 11 Set 𝜃 𝑧 = 𝑇 −1 𝑇𝑗=0 12 Compute 𝑓 (𝜃 𝑧 ) = 𝑔(𝑦(𝜃 𝑧 ))𝑒 −ℓ ( 𝜃𝑧 ) ; 13 Choose 𝑧ˆ ∈ argmax 𝑧 ∈ N 𝑓 (𝜃 𝑧 ) and return 𝜃ˆ = 𝜃 𝑧ˆ ; It remains to analyze the algorithm. Proof of Theorem 4.14. We analyze Algorithm 3. With its choices 𝜉 = 𝛿/6 and 𝜁 = 𝛿/2, Lemma 4.16 gives the desired error 3𝜉 + 𝜁 = 𝛿, provided each convex minimization attains error 𝜁. We verify this guarantee and the runtime below. √ Constructing the net. Identify 𝑊 with R𝑞 , where 𝑞 = 2 dimC 𝑊. For 𝑞 > 0, set 𝑚 = ⌈2 𝑞/𝑡⌉, take the grid {−1 + 2 𝑗/𝑚 : 0 ≤ 𝑗 ≤ 𝑚} 𝑞 , and project its points radially onto the unit ball. Every point in the ball is within 𝑡/2 of a grid point; radial projection moves that grid point by at most another 𝑡/2. This gives a 𝑡-net with √ |N | ≤ (𝑚 + 1) 𝑞 ≤ (2 + 2 𝑞/𝑡) 𝑞 . If 𝑊 = {0}, take N = {0}. Explicitly, the choices in the algorithm give   2 𝛿 6Γ 𝑡= , 𝑞 ≤ 𝑄 ℎ,𝐾 , 𝛿 := 576(𝐾 + 1) 2 , 36𝐾Γ 𝛿 ! 𝑄ℎ,𝐾, 𝛿 √︁ 72𝐾Γ 𝑄 ℎ,𝐾 , 𝛿 |N | ≤ 2 + . 𝛿 Applying √ the convex optimization black box. The domain Dℎ is compact and convex, contains 0, and has ∥𝜃 ∥ P ≤ 5ℎ. Lemma 4.16 establishes convexity of 𝐽𝑧 , which is also continuous. Proposition 4.18 supplies √ the required subgradients and projections with 𝐺 = 𝐺 ℎ + 𝜆/𝑅 ℎ . Thus Fact 4.17, applied with 𝐷 0 = 5ℎ and the 𝑇, 𝛼 in Algorithm 3, gives 𝐽𝑧 (𝜃 𝑧 ) ≤ min 𝐽𝑧 + 𝜁 . Dℎ

Every candidate therefore meets the premise of Lemma 4.16, using 𝑂 (𝑇) oracle calls each. Here the iteration

44

count is explicitly &

  2' 36𝐾Γ(𝐵 ℎ + 1) 20ℎ 𝐺ℎ + 𝑇= ≍ (ℎ + 1) 7 𝛿 −2 + (ℎ + 1) 4 𝐾 2 Γ2 𝛿 −4 . 𝑅ℎ 𝛿 𝛿2 Objective evaluation and total cost. Evaluate 𝑓 (𝜃 𝑧 ) = 𝑝(𝜃 𝑧 )𝑒 −ℓ ( 𝜃𝑧 ) directly using the finite quadratic form for 𝑝 and (11) for ℓ. Constructing the tensor and 𝑊, and evaluating each numerator, costs 𝐶 (𝐾 + 1) 𝐶 𝐾!(𝑛 + 1) 𝑐𝐾 operations. Each convex iteration has polynomial cost in 𝑛, 𝑞. Multiplying by the net-size and iteration bounds above, and absorbing their polynomial factors, gives the explicit prefactor (60) and hence 𝐶ℎ,𝐾 , 𝛿 (𝑛 + 1) 𝑐𝐾 operations for universal 𝑐, as claimed. □ Proof of Theorem 4.9. Choose a large enough 𝐾 by (55) and perform Lemma 4.13 with accuracy parameter 𝜖 and failure probability 𝜂. On its success event, (59) holds simultaneously for all 𝜃 ∈ Dℎ , so optimizing on the estimated density matrix in the subspace 𝑃𝐾 suffices to approximately optimize the true fidelity with 𝜎. Apply Theorem 4.14 with 𝛿 = 𝜖/2 to get 𝜋ˆ = 𝜋 𝜃ˆ , which we claim is the approximately maximizing pure Gaussian state. Every 𝜋 with 𝜈(𝜋) ≤ ℎ has parameters in Dℎ by Lemma 4.10, so ˆ − 𝜖/8 𝐹 (𝜎, 𝜋) ˆ ≥ 𝑓 ( 𝜃) ≥ max 𝑓 − 𝜖/2 − 𝜖/8 ≥ 𝐹 (𝜎, 𝜋) − 3𝜖/4. Dℎ

Taking the supremum over 𝜋 with bounded photon number proves the fidelity guarantee. Moreover, 𝜈( 𝜋) ˆ ≤ 3 17(ℎ + 1) . Tomography uses 𝑂 (𝐷 4𝐾 𝜖 −2 log(2𝐷 𝐾 /𝜂)) copies and polynomial classical processing. Since 𝐷 𝐾 ≤ (𝑛 + 1) 𝐾 , its copy count is at most 𝐶 (𝐾 + 1)𝜖 −2 (𝑛 + 1) 4𝐾+1 log(2/𝜂). An explicit choice of the prefactor is
𝐶ℎ, 𝜖 = 𝐶 (𝐾 ℎ, 𝜖 + 1)𝜖 −2 + 𝐶ℎ,𝐾ℎ, 𝜖 , 𝜖 /2 ,

(66)

where 𝐶 is universal and 𝐶ℎ,𝐾 , 𝛿 is specified in (60). The exponent of 𝑛 + 1 can be taken to be 𝑐 ∗ 𝐾 ℎ, 𝜖 for a sufficiently large universal 𝑐 ∗ . Taking 𝐾 = 𝐾 ℎ, 𝜖 and adding the optimization cost gives (51). □

4.3

Main result

Now we state our main result for the low-fidelity regime. The proof is by combining Theorem 4.1 and Therorem 4.9. Theorem 4.19 (Low-fidelity agnostic learning). Let 𝜌 be an arbitrary 𝑛-mode bosonic state. Let opt = inf 𝜋 ∈ G 𝑑IF (𝜌, 𝜋), let 𝜋∗ ∈ G attain this optimum value, and let 𝐸 0 ≥ 3 be an upper bound on 𝜈(𝜋∗ ). Given 𝜖 ∈ (0, 1] and 𝛿 ∈ (0, 1/2], there is an algorithm which outputs 𝜋ˆ ∈ G such that, with probability at least 1 − 𝛿, 𝑑IF (𝜌, 𝜋) ˆ ≤ opt + 𝜖, using at most 𝑛poly(1/𝜖 ) · poly log(1/𝛿), log log(𝐸 0 )

45

copies of 𝜌. The runtime of our algorithm is

exp exp(poly(1/𝜖)) · 𝑛poly(1/𝜖 ) · poly log(1/𝛿), log log 𝐸 0 . Proof. Set 𝛼 = 𝜖/4, 𝐴0 = 4𝐸 0 + 2, and 𝑊 = ⌈8/(3𝛼)⌉. Apply Theorem 4.1 with failure probability 𝛿/3. If it reports failure, return vacuum. Otherwise, for every returned 𝐺, apply Theorem 4.9 to 𝐺 𝜌𝐺 † with ℎ = 𝐻 𝛼 , accuracy 𝜖/4, and failure probability 𝛿/(3𝑊). Rotate the resulting state back by applying 𝐺 † to obtain 𝜋˜ 𝐺 . Estimate each 𝐹 (𝜌, 𝜋˜ 𝐺 ) to additive error 𝜖/4 and failure probability 𝛿/(3𝑊) on fresh copies, and return the candidate with the largest estimate. For any pure Gaussian candidate 𝜋˜ 𝐺 = 𝑈 |0⟩⟨0| 𝑈 † , applying 𝑈 † and testing vacuum has success probability 𝐹 (𝜌, 𝜋˜ 𝐺 ). Hoeffding’s inequality therefore gives the stated accuracy with 𝑂 (𝜖 −2 log(6𝑊/𝛿)) copies, without any proximity-to-vacuum assumption on 𝜌. If 𝐹 (𝜌, 𝜋∗ ) < 𝛼, every output satisfies 𝐹 (𝜌, 𝜋) ˆ ≥ 0 > 𝐹 (𝜌, 𝜋∗ ) − 𝜖. Otherwise, (4) gives 𝜅(𝑉∗ ) ≤ 𝐴0 . With probability at least 1− 𝛿, preprocessing, every optimizer call, and every validation succeed. Conditional failure bounds are valid because these routines use fresh copies. On this event a useful basis 𝐺 ∗ satisfies 𝜈(𝐺 ∗ 𝜋∗ 𝐺 †∗ ) ≤ 𝐻 𝛼 , and hence 𝐹 (𝜌, 𝜋˜ 𝐺∗ ) ≥ 𝐹 (𝜌, 𝜋∗ ) − 𝜖/4. Selecting the largest estimate loses at most two validation errors, so 𝐹 (𝜌, 𝜋) ˆ ≥ 𝐹 (𝜌, 𝜋˜ 𝐺∗ ) − 2(𝜖/4) ≥ 1 − opt − 3𝜖/4 ≥ 1 − opt − 𝜖 . This proves the desired infidelity guarantee. The preprocessing cost is (37), with 𝑐 𝛼 = 𝐶dec (16/𝜖) 𝑟dec and 𝐻 𝛼 given by (50). The 𝑊 = ⌈32/(3𝜖)⌉ optimizer calls use in total at most 𝑊𝐶 𝐻 𝛼 , 𝜖 /4 (𝑛 + 1) 𝑐∗ 𝐾𝐻 𝛼 , 𝜖 /4 log(6𝑊/𝛿) copies and arithmetic operations, using the constants from the proof of Theorem 4.9. Validation costs 𝑂 (𝑊𝜖 −2 log(6𝑊/𝛿)) copies. To record the parameter dependence, set 𝐾 𝜖 = 𝐾 𝐻 𝛼 , 𝜖 /4 , 𝑘 𝜖 = max{𝑐 𝛼 , 𝑐 ∗ 𝐾 𝜖 }, h i
𝐶 𝜖 = 𝐶 𝐶 𝛼 + 𝑊 𝐶 𝐻 𝛼 , 𝜖 /4 + 𝜖 −2 (1 + log(6𝑊)) . Here 𝐶 𝛼 , 𝑐 𝛼 are from (37), and 𝐾 ℎ,𝑎 , 𝐶ℎ,𝑎 , 𝑐 ∗ from (55) and (66). The total arithmetic cost is bounded by 𝐶 𝜖 (𝑛 + 1) 𝑘 𝜖 times a universal polynomial in 1 + log(1/𝛿) and 1 + log log(𝑒 + 𝐸 0 ). Since 𝐻 𝜖 /4 ≍ 𝜖 −𝑠 with 𝑠 ≍ 𝑟 dec , we have    𝐾 𝜖 ≍ 𝑘 𝜖 ≍ 𝜖 −3𝑠 , 𝐶 𝜖 ≤ exp exp 𝐶𝜖 −3𝑠 log(2/𝜖) . For the copy count, the 𝑊 optimizer calls only require the tomography cost from Lemma 4.13, totaling 𝐶𝑊 (𝐾 𝜖 + 1)𝜖 −2 (𝑛 + 1) 4𝐾 𝜖 +1 log(6𝑊/𝛿). Adding preprocessing and validation, and absorbing their exp(poly(1/𝜖)) factors into (𝑛 + 1) poly(1/𝜖 ) , gives the stated copy and decoder-call bounds. The net search contributes only arithmetic operations. These choices have no dependence on opt; it enters only the target fidelity 1 − opt − 𝜖. Substituting 𝐴0 = 4𝐸 0 + 2

46

proves the complexity bounds in Theorem 4.19.

5

□

Agnostic learning with Gaussian measurements

In Section 3, we showed that learning in the high-fidelity regime is naively possible to 𝑂˜ (opt) + 𝜖 error by embedding robust estimators into the optimal tomography protocol of [CMF+ 26]; we then gave an algorithm to iteratively refine this warm start estimate to opt + 𝜖 error using on-off photon detection or alternatively use more complicated non-Gaussian measurements which are harder to experimentally implement but enable us to achieve the same sample complexity as tomography itself, up to log factors. This leaves us, however, with two questions: Are these non-Gaussian measurements necessary? Can we do better than 𝑂˜ (opt) + 𝜖 using only Gaussian measurements? In this section, we answer both in the affirmative, finding that Gaussian measurements are capable of refining the warm start estimate to 𝑂 (opt) + 𝜖 error but that opt + 𝜖 is only efficiently achievable using non-Gaussian measurements.

5.1

Refining to 𝑂 (opt) + 𝜖 with Gaussian measurements

We first state our formal guarantee for the task of agnostically learning pure bosonic Gaussian states in infidelity, using only Gaussian measurements. Theorem 5.1. Let 𝜌 be an 𝑛-mode state where 𝑑IF (𝜌, |𝜓⟩⟨𝜓|) ≤ opt ≤ opt for pure Gaussian state |𝜓⟩ with covariance 𝑉. Then, given 𝜖 > 0 and 𝛿 ∈ (0, 1/2), there exists an algorithm takes 𝑂˜ (𝑛2 𝜖 −1 log log ∥𝑉 −1 ∥ op log5 (1/𝛿)) ˆ satisfying copies of 𝜌 and uses only Gaussian measurements to output a pure Gaussian state | 𝜓⟩ ˆ 𝜓|) ˆ ≤ 𝑂 (opt) + 𝜖 𝑑IF (𝜌, | 𝜓⟩⟨ with probability at least 1 − 𝛿. Proof. We achieve this by making a petite modification to the final estimation step of the naive warm start algorithm which proceeds by iteratively unsqueezing the state, estimating it via robust heterodyne tomography, and then projecting the estimate to a pure Gaussian state. As before, WLOG, we ignore the unsqueezing steps and assume ∥𝑉 −1 ∥ op ≤ 2 so that we may use the notation of the original state for simplicity along with the fact that 21 𝐼 ⪯ 𝑉 ⪯ 2𝐼. √ Now, recall that in the final estimation step of the naive warm start algorithm, we use 𝑑tr (𝜌, |𝜓⟩⟨𝜓|) ≤ opt √ from Fuchs-van de Graaf to view the outcome distribution, 𝑃, of heterodyning 𝜌 as being opt-close in total variation distance to 𝑄 ≡ N (𝜇 ℎ , Σℎ ) such that we can apply the robust Gaussian learner in Proposition 2.9 to √ learn (𝜇 ℎ , Σℎ ) to absolute Frobenius error 𝑂˜ ( opt). By Lemma 3.7, this means that the pure state projection √ is 𝑂˜ ( opt)-close in trace distance to |𝜓⟩⟨𝜓|, and by the manipulation in (21), this gives an estimate which is 𝑂˜ (opt)-close in infidelity to 𝜌. Thus, by the same machinery, proving Theorem 5.1 can be reduced to √ learning (𝜇 ℎ , Σℎ ) to absolute Frobenius error 𝑂 ( opt) from the heterodyne outcomes. We begin by introducing a different classical robust estimation primitive.

47

Proposition 5.2 (Proposition 1.5 of [DKP20]). Let D be a distribution over R𝑑 with mean 𝜇 and bounded covariance Σ ⪯ 𝐵𝐼 for some known constant 𝐵 > 0. Then, there exists a polytime algorithm which given 𝑁 = 𝑂˜ (𝜖 −2 𝑑 log(1/𝛿)) 𝜂-corrupted samples from D outputs 𝜇ˆ such that √ ∥ 𝜇ˆ − 𝜇∥ 2 ≤ 𝑂 ( 𝜂) + 𝜖

(67)

with probability at least 1 − 𝛿. √ At first glance, Proposition 5.2 seems worse than Proposition 2.9: 𝑑tv (𝑃, 𝑄) ≲ opt means that taking D = 𝑄 √ and 𝜂 = opt gives only 𝑂 (opt1/4 ) absolute error in estimating the mean of the heterodyne distribution with this learner. However, unlike Proposition 2.9, Proposition 5.2 does not require D to be Gaussian. The key insight is then to write 𝑃 as some 𝑂 (opt)-corruption of a distribution D where instead of taking D = 𝑄, we √ allow D to be a non-Gaussian distribution but require the mean of D to be 𝑂 ( opt)-close to the mean of 𝑄. √ Then, by triangle inequality and the guarantee in (67), the estimated mean is 𝑂 ( opt)-close to the mean of D and also the mean of 𝑄. √ We first recall that 𝑑IF (𝜌, |𝜓⟩⟨𝜓|) ≤ opt implies not just opt-closeness in total variation distance but also opt-closeness in squared Hellinger distance: ∫ √︁ √︁ √︁ 2 𝑑 𝐻 (𝑃, 𝑄) = 1 − 𝑝(𝑥)𝑞(𝑥) 𝑑𝑥 ≤ 1 − 𝐹 (𝜌, |𝜓⟩⟨𝜓|) ≤ 1 − 1 − opt ≤ opt (68) where the first inequality uses √ the monotonicity of fidelity under measurement and where the last inequality uses the elementary fact 1 − 1 − 𝑥 ≤ 𝑥 on [0, 1]. The main technical lemma is then the following conversion from Hellinger closeness to the target Gaussian and Huber closeness to a perhaps non-Gaussian distribution that is close in mean to the original target distribution. Lemma 5.3. Let 𝑃, 𝑄 be distributions over R𝑑 such that 𝑑 2𝐻 (𝑃, 𝑄) < ℎ for universal constant ℎ, with 𝑄 having mean 𝜇 and bounded covariance Σ ⪯ 𝐵𝐼 for known constant 𝐵 > 0. Then, there exists 𝜂 ≲ 𝑑 2𝐻 (𝑃, 𝑄) and distributions D, R such that 𝑃 = (1 − 𝜂)D + 𝜂𝑅,

∥𝜇 D − 𝜇∥ 2 ≲ 𝑑 𝐻 (𝑃, 𝑄),

∥Σ D ∥ op ≤ 𝐵′

(69)

where 𝜇 D , Σ D are the mean and covariance of D and 𝐵′ is a known function of 𝐵. Proof. We form D by taking 𝑃 and removing portions where the density significantly exceeds the density of 𝑄. Specifically, fixing an 𝛼 ≥ 4, let 𝑆 = {𝑥 : 𝑝(𝑥) > 𝛼𝑞(𝑥)} be the subset of the support we will remove, and decompose 𝑃 = (1 − 𝜂)D + 𝜂𝑅 with D = 𝑃(·|𝑆 𝑐 ), R = 𝑃(·|𝑆), and 𝜂 = Pr 𝑃 [𝑥 ∈ 𝑆]. Then, ∫ √︁ ∫ √︁ 𝜂= 𝑝(𝑥)𝑑𝑥 ≤ 𝛼 ( 𝑝(𝑥) − 𝑞(𝑥)) 2 𝑑𝑥 ≤ 2𝛼𝑑 2𝐻 (𝑃, 𝑄) (70) 𝑆

𝑆

√︁

√︁ where the first inequality uses the fact that ( 𝑝(𝑥) − 𝑞(𝑥)) 2 ≥ (1 − 𝛼 −1/2 ) 2 𝑝(𝑥) ≥ 𝑝(𝑥)/𝛼 on 𝑆 for 𝛼 ≥ 4 and where the second inequality uses the definition of Hellinger distance. Now, set ℎ = 1/(4𝛼) such that 𝜂 ≤ 1/2.

48

We now bound the difference in means. Let 𝑢 ∈ R𝑑 be a unit vector. Then, ∫ ∫ ∫ 1 1 |⟨𝑢, 𝜇 D − 𝜇⟩| = ⟨𝑢, 𝑥 − 𝜇⟩ 𝑝(𝑥)𝑑𝑥 = ⟨𝑢, 𝑥 − 𝜇⟩( 𝑝(𝑥) − 𝑞(𝑥))𝑑𝑥 − ⟨𝑢, 𝑥 − 𝜇⟩𝑞(𝑥)𝑑𝑥 1 − 𝜂 𝑆𝑐 1 − 𝜂 𝑆𝑐 𝑆

≤

1 1−𝜂 +

 1/2 ∫ √︁  1/2 √︁ √︁ √︁ ⟨𝑢, 𝑥 − 𝜇⟩ 2 ( 𝑝(𝑥) + 𝑞(𝑥)) 2 𝑑𝑥 ( 𝑝(𝑥) − 𝑞(𝑥)) 2 𝑑𝑥

∫

1 1−𝜂

𝑆𝑐

𝑆𝑐

∫

⟨𝑢, 𝑥 − 𝜇⟩ 2 𝑞(𝑥)𝑑𝑥

𝑆

 1/2 ∫

 1/2 𝑞(𝑥)𝑑𝑥

𝑆

∫  1/2 √ √ 1 √︂ 1 2 · ( 𝛼 + 1) ⟨𝑢, 𝑥 − 𝜇⟩ 𝑞(𝑥)𝑑𝑥 · 2𝑑 𝐻 (𝑃, 𝑄) + 𝐵 Pr [𝑥 ∈ 𝑆] ≤ 𝑄 1−𝜂 1−𝜂 𝑆𝑐 √   √︁ √ √ 𝐵 ( 𝛼 + 1) 2𝑑 𝐻 (𝑃, 𝑄) + 𝜂/𝛼 ≤ 1−𝜂 ≲ 𝑑 𝐻 (𝑃, 𝑄) ∫ where the first line uses the fact that R𝑑 ⟨𝑢, 𝑥 − 𝜇⟩𝑞(𝑥)𝑑𝑥 = 0, the second line uses triangle inequality and Cauchy-Schwarz, and the final line uses 𝜂 ≲ 𝑑 2𝐻 (𝑃, 𝑄) from (70). Since this holds for any unit vector 𝑢 ∈ R𝑑 , by the variational characterization of the ℓ2 norm, ∥𝜇 D − 𝜇∥ 2 ≲ 𝑑 𝐻 (𝑃, 𝑄) as desired. We finally show that that D has bounded covariance as well. For unit vector 𝑢, 𝑢 ⊤ Σ D 𝑢 = E [⟨𝑢, 𝑥 − 𝜇 D ⟩ 2 ] ≤ E [⟨𝑢, 𝑥 − 𝜇⟩ 2 ] ≤ D

D

𝛼 E [⟨𝑢, 𝑥 − 𝜇⟩ 2 ] ≤ 2𝛼𝐵 ≡ 𝐵′ (𝐵) 1−𝜂 𝑄

where the first inequality uses the fact that the mean minimizes the MSE and where the second inequality uses 1 [ 𝑥 ∈𝑆 𝑐 ] the fact that 𝑝 D (𝑥) = 𝑝 ( 𝑥 )1− ≤ 1−𝛼𝜂 𝑞(𝑥). Then, taking the supremum over all such 𝑢, ∥Σ D ∥ op ≤ 𝐵′ (𝐵) 𝜂 as well. □ Collating Lemma 5.3 and Proposition 5.2 together gives the following mean estimator for bounded covariance distributions that is robust to Hellinger misspecification. Corollary 5.4. Let 𝑃, 𝑄 be distributions over R𝑑 such that 𝑑 2𝐻 (𝑃, 𝑄) ≤ 𝜂 ≤ 𝜂 for some universal constant 𝜂 > 0, with 𝑄 having mean 𝜇 and bounded covariance Σ ⪯ 𝐵𝐼 for known constant 𝐵 > 0. Then, there exists a polytime algorithm which given 𝑁 = 𝑂˜ (𝜖 −2 𝑑 log(1/𝛿)) samples from 𝑃 outputs 𝜇ˆ such that √ ∥ 𝜇ˆ − 𝜇∥ 2 ≤ 𝑂 ( 𝜂) + 𝜖 with probability at least 1 − 𝛿. Proof. Applying Lemma 5.3, Proposition 5.2 and identifying D, we have a 𝜇ˆ such that: √ √ √ ∥ 𝜇ˆ − 𝜇∥ 2 ≤ ∥ 𝜇ˆ − 𝜇 D ∥ 2 + ∥𝜇 D − 𝜇∥ 2 ≤ (𝑂 ( 𝜂) + 𝜖) + 𝜂 = 𝑂 ( 𝜂) + 𝜖 where the first term is from (67) and the second term is from (69). When 𝑄 is Gaussian, we can also learn the covariance with a touch more work.

49

□

Corollary 5.5. Let 𝑃, 𝑄 be distributions over R𝑑 such that 𝑑 2𝐻 (𝑃, 𝑄) ≤ 𝜂 ≤ 𝜂 for some universal constant 𝜂 > 0, with 𝑄 = N (𝜇, Σ) having bounded covariance Σ ⪯ 𝐵𝐼 for known constant 𝐵 > 0. Then, there exists a polytime algorithm which given 𝑁 = 𝑂˜ (𝜖 −2 𝑑 2 log(1/𝛿)) samples from 𝑃 outputs 𝜇, ˆ Σ̂ such that √ ∥ 𝜇ˆ − 𝜇∥ 2 ≤ 𝑂 ( 𝜂) + 𝜖,

√ ∥ Σ̂ − Σ∥ 𝐹 ≤ 𝑂 ( 𝜂) + 𝜖

with probability at least 1 − 𝛿. √ Proof. Corollary 5.4 uses 𝑂˜ (𝜖1−2 𝑑 log(1/𝛿)) samples to give a 𝜇ˆ such that ∥Δ∥ 2 ≤ 𝑂 ( 𝜂) + 𝜖1 , where Δ ≡ 𝜇ˆ − 𝜇. Now, taking fresh samples 𝑋𝑖 , we form 𝑌𝑖 = (𝑋𝑖 − 𝜇) ˆ (𝑋𝑖 − 𝜇) ˆ ⊤ , which has the distribution 𝑄𝑌 iid iid or 𝑃𝑌 if 𝑋𝑖 ∼ 𝑄 or 𝑋𝑖 ∼ 𝑃 respectively. Viewing 𝑌𝑖 as a vector such that the ℓ2 guarantee corresponds with √ the Frobenius norm, we claim that Corollary 5.4 gives an estimate Σ̂ with ∥ Σ̂ − Σ∥ 𝐹 ≤ 𝑂 ( 𝜂) + 𝜖. First, by data processing, we have that 𝑑 2𝐻 (𝑃𝑌 , 𝑄𝑌 ) ≤ 𝑑 2𝐻 (𝑃, 𝑄) ≤ 𝜂 iid

as desired. Then, we can view 𝑋𝑖 = 𝜇 + 𝑍 for 𝑍𝑖 ∼ N (0, Σ), meaning E𝑄𝑌 [𝑌 ] = E[(𝑍 − Δ) (𝑍 − Δ) ⊤ ] = Σ + ΔΔ⊤ . Assuming for now that the conditions of Corollary 5.4 are valid, √ √ √ √ ∥ Σ̂ − Σ∥ 𝐹 ≤ ∥ Σ̂ − E [𝑌 ] ∥ 𝐹 + ∥ΔΔ⊤ ∥ 𝐹 ≤ (𝑂 ( 𝜂) + 𝜖 1 ) + ∥Δ∥ 22 ≤ 𝑂 ( 𝜂) + 𝜖 1 + (𝑂 ( 𝜂) + 𝜖 1 ) 2 ≤ 𝑂 ( 𝜂) + 𝜖 𝑄𝑌

where both calls of Corollary 5.4 use slack 𝜖 1 , which we define such that the final slack is 𝜖. Thus, it suffices to show that 𝑄𝑌 has bounded covariance. We do this by showing that Var𝑄𝑌 (⟨𝑀, 𝑌 ⟩𝐹 ) ≤ 𝐵 for all 𝑀 ∈ R𝑑×𝑑 with ∥𝑀 ∥ 𝐹 = 1, for some 𝐵 > 0. Since 𝑌 is symmetric, it suffices to consider symmetric 𝑀. Then, by standard moment identities regarding multivariate Gaussians, specifically (379) of [PP+ 08], ∥ 2𝐹 ∥𝑀 Var𝑄𝑌 (⟨𝑀, 𝑌 ⟩𝐹 ) = Var𝑄𝑌 ((𝑍 − Δ) ⊤ 𝑀 (𝑍 − Δ)) = 2 tr(𝑀Σ𝑀Σ) + 4Δ⊤ 𝑀Σ𝑀Δ ≤ (2𝐵2 + 4𝐵∥Δ∥ 22 )



≤ 2𝐵2 + 4𝐵 where we use the bounds Σ ⪯ 𝐵𝐼 and ∥Δ∥ 22 ≤ 1 in the final two inequalities. Thus, 𝑄𝑌 has bounded covariance, meaning we are done. □ Thus, (68) and Corollary 5.5 give estimators ( 𝜇ˆ ℎ , Σ̂ℎ ) that resolve (𝜇 ℎ , Σℎ ) to additive absolute error √ 𝑂 ( opt) + 𝜖, meaning we are done. □

5.2

Hardness for Gaussian measurements

We now show that non-Gaussian measurements are info-theoretically necessary for achieving opt + 𝜖 error. Theorem 5.6. For universal constant 𝑐 > 0, sufficiently large 𝑛, and 𝜂 ∈ (0, 1/4], there exist two (𝑛+1)-mode states 𝜌+ , 𝜌 − satisfying opt+ = opt− = 𝑂 (𝜂), 𝑑tr (𝜌± , |0⟩ ⊗ (𝑛+1) ) ≤ 𝜂

50

where opt± ≡ inf | 𝜓⟩ ∈ G 𝑑IF (𝜌± , |𝜓⟩), such that any algorithm using only Gaussian measurements that outputs | 𝜓ˆ ± ⟩ with the guarantee 𝑑IF (𝜌± , | 𝜓ˆ ± ⟩) ≤ opt± + 𝑐𝜂2 with probability at least 2/3 must have copy complexity
𝑁 ≥ 𝜂 −1 exp Ω(𝑛1/3 ) . Since Gaussian measurements cannot distinguish states better than their Wigner functions allow, we construct 𝜌+ , 𝜌 − with extremely close Wigner functions for which the learner is forced to output different pure Gaussian states. We do this by first constructing two Wigner-close separable bosonic states 𝜎+ , 𝜎− whose vacuum probabilities differ significantly and then coupling this difference to a coherent direction in a new mode. We start by importing the following moment matching construction. Lemma 5.7. For every integer 𝑘 ≥ 1, there exist random variables 𝑈+ , 𝑈− supported on {0} ∪ [1, 𝐶 𝑘 2 ] with the following properties: E[𝑈± ] = 1,

𝑗

E[𝑈+ ] = E[𝑈−𝑗 ]

( 𝑗 ∈ [𝑘]),

Pr[𝑈+ = 0] − Pr[𝑈− = 0] ≥ 𝑐

where 𝐶, 𝑐 > 0 are universal constants. We defer the proof to Appendix A.4.1. We now use 𝑈+ , 𝑈− to construct 𝜎+ , 𝜎− such that the moment matching gives Wigner-closeness while the difference in probability mass on zero gives a significant difference in vacuum probability. Lemma 5.8. There exists 𝑐 > 0 such that for every sufficiently large 𝑛, there exist 𝑛-mode states 𝜎+ , 𝜎− that are separable, diagonal in the Fock basis and have nonnegative Wigner functions such that:   ⟨0| ⊗𝑛 𝜎+ |0⟩ ⊗𝑛 − ⟨0| ⊗𝑛 𝜎− |0⟩ ⊗𝑛 ≥ 𝑐, 𝑑tv (𝑊 𝜎+ , 𝑊 𝜎− ) ≤ exp −Ω(𝑛1/3 ) (71) Proof. Choosing 𝑘 = 𝑐 𝑘 𝑛1/3 for suitably small constant 𝑐 𝑘 > 0, define the following mixed state:   ⊗𝑛  𝑠𝑢  𝑠𝑢 |0⟩⟨0| + |1⟩⟨1| 𝜎± = E 1− 𝑢∼𝑈± 2𝑛 2𝑛 for suitably large scaling constant 𝑠, where we take 𝑛 large enough such that 𝑠𝑢/(2𝑛) < 1/2. Considering the vacuum probabilities, h h 𝑠𝑢  𝑛 i 𝑠𝑢  𝑛 i ⟨0| ⊗𝑛 𝜎+ |0⟩ ⊗𝑛 − ⟨0| ⊗𝑛 𝜎− |0⟩ ⊗𝑛 = E 1− − E 1− 𝑢∼𝑈+ 𝑢∼𝑈− 2𝑛 2𝑛 −𝑠/2 ≥ Pr[𝑈+ = 0] − (Pr[𝑈− = 0] + 𝑒 Pr[𝑈− > 0]) ≥ 𝑐 − 𝑒 −𝑠/2 where we use the fact that (1 − 𝑠𝑢/(2𝑛)) 𝑛 ≤ 𝑒 −𝑠𝑢/2 ≤ 𝑒 −𝑠/2 when 𝑢 ≥ 1. Taking suitable 𝑐, 𝑠, the vacuum probability is bounded away from 0. We now show that the Wigner functions are close in total variation distance. Defining 𝑔(𝑞, 𝑝) = 2 2 −1 𝜋 𝑒 −𝑞 − 𝑝 , it is a known fact that the vacuum and one-photon Wigner functions are given by 𝑊 |0⟩⟨0| = 𝑔 and 𝑊 |1⟩⟨1| = (2𝑞 2 + 2𝑝 2 − 1)𝑔 respectively, where the distribution of 𝑞 2 + 𝑝 2 under joint density 𝑔(𝑞, 𝑝) can be viewed as an exponential random variable of unit mean. Using these facts and linearity, the 𝑖-th mode

51

of 𝜎± has Wigner function:    𝑠𝑢 𝑠𝑢 𝑠𝑢  𝑊 ( 1− 𝑠𝑢 ) |0⟩⟨0|+ 𝑠𝑢 |1⟩⟨1| = 𝑔(𝑞 𝑖 , 𝑝 𝑖 ) 1 − + (2𝑞 2𝑖 + 2𝑝 2𝑖 − 1) = 𝑔(𝑞 𝑖 , 𝑝 𝑖 ) 1 + 𝑍𝑖 2𝑛 2𝑛 2𝑛 2𝑛 𝑛 where 𝑍𝑖 = 𝑞 2𝑖 + 𝑝 2𝑖 − 1 gives E𝑔𝑖 [𝑍𝑖 ] = 0 and E𝑔𝑖 [𝑍𝑖2 ] = 1. Then, by linearity and the tensor-product rule for the Wigner function, # # " 𝑛 " 𝑛  Ö Ö 𝑠𝑢  𝑠𝑢  ⊗𝑛 E 1 + 𝑍𝑖 (72) 𝑊 𝜎± = E 𝑔(𝑞 𝑖 , 𝑝 𝑖 ) 1 + 𝑍𝑖 = 𝑔 𝑢∼𝑈± 𝑢∼𝑈± 𝑛 𝑛 𝑖=1 𝑖=1 where 𝑔 ⊗𝑛 (𝑞, 𝑝) =

Î𝑛

𝑖=1 𝑔(𝑞 𝑖 , 𝑝 𝑖 ) is the natural product density. By moment matching,

𝑊 𝜎+ − 𝑊 𝜎− = 𝑔 ⊗𝑛

∑︁ 𝑆 ⊆ [𝑛]:|𝑆 |>𝑘

 𝑠  |𝑆 |  𝑛

E[𝑈+|𝑆 | ] − E[𝑈−|𝑆 | ]

Ö

𝑍𝑖

𝑖 ∈𝑆

Then, by Cauchy-Schwarz, ∥𝑊 𝜎+ − 𝑊 𝜎− ∥ 21 "   #  𝑊 𝜎+ − 𝑊 𝜎− 2 𝑊 𝜎+ − 𝑊 𝜎− 2 ≤ E⊗𝑛 = E⊗𝑛 𝑔 ⊗𝑛 𝑔 ⊗𝑛 𝑔 𝑔  𝑠  |𝑆1 |+|𝑆2 | 

∑︁

=

𝑆1 ,𝑆2 ⊆ [𝑛]:|𝑆1 |>𝑘, |𝑆2 |>𝑘

𝑛

E[𝑈+|𝑆1 | ] − E[𝑈−|𝑆1 | ]



E[𝑈+|𝑆2 | ] − E[𝑈−|𝑆2 | ]

E

𝑔 ⊗𝑛

| =

𝑛    2 𝑗 ∑︁ 𝑛 𝑠

𝑗 𝑗=𝑘+1 𝑛 ∑︁

≤4

𝑛

#

"



Ö

𝑍𝑖

𝑖 ∈𝑆1

{z

Ö

𝑍𝑖

𝑖 ∈𝑆2

1 [𝑆1 =𝑆2 ]

}

(E[𝑈+ ] − E[𝑈−𝑗 ]) 2 𝑗

(𝑒𝐶 2 𝑠2 𝑘 3 /𝑛) 𝑗

𝑗=𝑘+1

where we use the independence, zero mean, and unit variance properties of 𝑍𝑖 as well as boundedness
𝑈± ∈ [0, 𝐶 𝑘 2 ] and the standard fact 𝑛𝑗 ≤ 𝑛 𝑗 / 𝑗! ≤ (𝑒𝑛/ 𝑗) 𝑗 . Then, if 𝑘 = 𝑐 𝑘 𝑛1/3 for suitably small 𝑐 𝑘 , we can bound 𝑒𝐶 2 𝑠2 𝑘 3 /𝑛 ≤ 1/4, giving ∥𝑊 𝜎+ − 𝑊 𝜎− ∥ 21 ≤ 𝑂 (4−𝑘 ) by bounding the tail sum of a geometric series. Since 𝑊 𝜎± ≥ 0 from (72), 𝑑tv (𝑊 𝜎+ , 𝑊 𝜎− ) ≤ exp(−Ω(𝑘)) which proves (71). □ We now force our agnostic learner to distinguish this difference in vacuum probability by placing the difference along a coherent direction. Specifically, we define: 𝜌± = (1 − 𝜂) |0⟩⟨0| ⊗ (𝑛+1) +

𝜂 (𝜎+ ⊗ 𝜔± + 𝜎− ⊗ 𝜔∓ ) 2

(73)

where 𝜔± = 43 |0⟩⟨0| + 1/4 |1⟩⟨1| ± 14 (|0⟩⟨1| + |1⟩⟨0|) which we can verify has 𝑊 𝜔± ≥ 0, meaning 𝑊𝜌± ≥ 0 by linearity such that ∥𝑊𝜌± ∥ 1 = 1 as well. By construction, these states are 𝜂-close to vacuum and differ by a Gaussian phase rotation on the last mode that maps 𝜔+ ↦→ 𝜔 − and vice versa, meaning opt+ = opt− = 𝑂 (𝜂).

52

We observe that the difference factors as 𝜌+ − 𝜌 − = 𝜂2 (𝜎+ − 𝜎− ) ⊗ (𝜔+ − 𝜔 − ). Then, by the tensor-product rule for Wigner functions,   𝜂 ∥𝑊𝜌+ − 𝑊𝜌− ∥ 1 = ∥𝑊 𝜎+ − 𝑊 𝜎− ∥ 1 ∥𝑊 𝜔+ − 𝑊 𝜔− ∥ 1 ≲ 𝜂 exp −Ω(𝑛1/3 ) 2 If 𝑃+ , 𝑃 − are the distributions of the learner’s output on 𝜌+ and 𝜌 − , by Lemma 3 of [SW21], 𝑑tv (𝑃+ , 𝑃 − ) ≤

  1 𝑁 ∥𝑊𝜌⊗+𝑁 − 𝑊𝜌⊗−𝑁 ∥ 1 ≤ ∥𝑊𝜌+ − 𝑊𝜌− ∥ 1 ≤ 𝑁𝜂 exp −Ω(𝑛1/3 ) 2 2

(74)

which establishes that no protocol using only Gaussian measurements, including those that use adaptivity or arbitrary classical processing, can distinguish 𝜌+ from 𝜌 − . Thus, Theorem 5.6 holds unless | 𝜓ˆ + ⟩ = | 𝜓ˆ − ⟩. We next show that the guarantees for 𝜌+ and 𝜌 − can not simultaneously be satisfied. Suppose there exists a |𝜓⟩ such that 𝑑IF (𝜌± , |𝜓⟩) ≤ opt+ + 𝑐𝜂2 . Then, if 𝜌¯ ≡ 21 (𝜌+ + 𝜌 − ), ⟨𝜓| 𝜌¯ |𝜓⟩ ≥ 1 − opt+ − 𝑐𝜂2

(75)

We now show the following lemma to derive a contradiction. Lemma 5.9. 1 − opt± > ∥ 𝜌∥ ¯ op + Ω(𝜂2 ). Proof. From (73), 𝜌¯ = (1 − 𝜂) |0⟩⟨0|

⊗ (𝑛+1)

  𝜂 3 1 |0⟩⟨0| + |1⟩⟨1| + (𝜎+ + 𝜎− ) ⊗ 2 4 4

which is diagonal in the Fock basis. For 𝜂 ≤ 1/4, since ∥𝜎± ∥ op ≤ 1 and ⟨0| 𝜔+ |0⟩ = ⟨0| 𝜔 − |0⟩, it follows that ∥ 𝜌∥ ¯ op = ⟨0| ⊗ (𝑛+1) 𝜌¯ |0⟩ ⊗ (𝑛+1) = ⟨0| ⊗ (𝑛+1) 𝜌± |0⟩ ⊗ (𝑛+1)

(76)

We now exhibit a pure Gaussian state that achieves a fidelity that surpasses this vacuum fidelity by Ω(𝜂2 ). Using the fact that ⟨0| 𝜔± |1⟩ = ±1/4 and ⟨1| 𝜔± |1⟩ = 1/4, consider the following elements of 𝜌± : 𝜂 (⟨0| ⊗𝑛 ⊗ ⟨0|) 𝜌± (|0⟩ ⊗𝑛 ⊗ |1⟩) = ± ⟨0| ⊗𝑛 (𝜎+ − 𝜎− ) |0⟩ ⊗𝑛 ≡ ±𝑡 8 2

We then consider the state |𝜙± ⟩ = |0⟩ ⊗𝑛 ⊗ |𝜓(±𝑡, 0)⟩ where |𝜓(±𝑡, 0)⟩ = 𝑒 −𝑡 /2

Í

𝑗 ≥0

(77) (±𝑡 ) 𝑗 √ | 𝑗⟩ from (5) is a 𝑗!

coherent state with this displacement on the last mode. Then,  −𝑡 2 ⟨0| ⊗ (𝑛+1) 𝜌± |0⟩ ⊗ (𝑛+1) + 2(±𝑡) (⟨0| ⊗𝑛 ⊗ ⟨0|) 𝜌± (|0⟩ ⊗𝑛 ⊗ |1⟩) 1 − opt± ≥ ⟨𝜙± | 𝜌± |𝜙± ⟩ = 𝑒  ⊗𝑛 ⊗𝑛 2 + 𝑡 (⟨0| ⊗ ⟨1|) 𝜌± (|0⟩ ⊗ |1⟩) 2

≥ 𝑒 −𝑡 (∥ 𝜌∥ ¯ op + 2𝑡 2 ) ≥ ∥ 𝜌∥ ¯ op + 𝑡 2 − 2𝑡 4

53

2

¯ op ≤ 1. Since 𝑡 = Θ(𝜂) where we drop the last term, recall (76), and use the fact that 𝑒 −𝑡 ≥ 1 − 𝑡 2 and ∥ 𝜌∥ 2 4 2 2 by Lemma 71 and (77), we can take 𝑡 ≤ 1/2 such that 𝑡 − 2𝑡 ≥ 𝑡 /2 > Ω(𝜂 ), proving the claim. □ Then, (75) and Lemma 5.9 imply ⟨𝜓| 𝜌¯ |𝜓⟩ ≥ 1 − opt+ − 𝑐𝜂2 > ∥ 𝜌∥ ¯ op for suitable 𝑐 > 0, which is a contradiction. This proves that no |𝜓⟩ can satisfy both the guarantee for 𝜌+ and that for 𝜌 − . It follows that 𝑑tv (𝑃+ , 𝑃 − ) ≥ 1/3, which together with (74) proves Theorem 5.6.

6

Hardness in the low-fidelity regime

For pure product states in the qubit setting, [BBK+ 25] show that a polytime agnostic learner in the low-fidelity regime can be used to efficiently approximate the spectral norm of a four-tensor 𝑇 ∈ C𝑚×𝑚×𝑚×𝑚 to additive Í 𝑚 are error. For ∥𝑇 ∥ 𝐹 = 1, they do this by first encoding 𝑇 ↦→ |𝜓𝑇 ⟩ ≡ 𝑖 𝑗 𝑘ℓ 𝑇𝑖 𝑗 𝑘ℓ 𝑢 𝑖 𝑢 𝑗 𝑢 𝑘 𝑢 ℓ where {|𝑢 𝑖 ⟩}𝑖=1 a Haar random choice of orthogonal unit vectors in span{|100 . . . 00⟩ , |010 . . . 00⟩ , . . . |000 . . . 01⟩} ⊂ C𝑛 , i.e. the subspace of single excitations. Then, taking 𝑛 = poly(𝑚/𝜖), they argue that the fidelity achieved by the best product state approximates some known constant multiple of the spectral norm of 𝑇 to additive error 𝜖. The hardness of this tensor problem comes from a neat fact in graph theory due to [MS65] which relates the following maximization problem over a graph, 𝐺, to the size of the maximum clique 𝜔(𝐺). Fact 6.1 (Theorem 1 of [MS65]). Let 𝐺 = ( [𝑛], 𝐸) be a graph with vertices labeled by the integers. Then,   ∑︁ ∑︁ 1 1 2 2 max |𝑥𝑖 | |𝑥 𝑗 | = max 𝑝𝑖 𝑝 𝑗 = 1− (78) 2 𝜔(𝐺) ∥ 𝑥 ∥ 2 =1 Í 𝑝𝑖 ≥0 {𝑖, 𝑗 } ∈𝐸

𝑖 𝑝𝑖 =1

{𝑖, 𝑗 } ∈𝐸

where the maximizer assigns weight 1/𝜔(𝐺) to each vertex of a maximum clique. In particular, Theorem 8.4 of [FL18] massages the left hand side of (78) into the spectral norm of a cleverly constructed four-tensor such that efficiently approximating this norm to inverse-polynomial error resolves a unique integer for 𝜔(𝐺), an NP-hard task. 𝑚 being single-photon Now, if we employ a similar strategy for pure Gaussian states with the {|𝑢 𝑖 ⟩}𝑖=1 excitations, squeezing, i.e. the quadratic terms in (5), cause |⟨𝜓(𝑏, 𝐴)|𝜓𝑇 ⟩| 2 to contain significant terms with couplings that do not appear in the spectral norm of 𝑇. Nevertheless, these can be controlled by instead Í encoding each index as coherent superpositions of pairs of photons |𝑢 𝑖 ⟩ = √1𝑟 𝑟𝑗=1 𝑎ˆ †𝐿 (𝑖, 𝑗 ) 𝑎ˆ †𝑅 (𝑖, 𝑗 ) |𝑎⟩ with a large 𝑟 = poly(1/𝜖) such that the optimum fidelity between |𝜓𝑇 ⟩ and a pure Gaussian state is exactly the spectral norm of 𝑇 up to additive error 𝜖. While one can show that this approach gives the desired lower bound, expanding and bounding the fidelity in terms of the Bargmann parameters is a bit arduous. Instead, this section gives a more straightforward proof that uses (78) to directly solve maximum clique with the claimed polytime agnostic learner. Theorem 6.2. Unless NP ⊆ BQP, there does not exist a quantum algorithm which takes any 𝜖 > 0, copies ˆ such that 𝑑IF (𝜌, | 𝜓⟩⟨ ˆ 𝜓|) ˆ ≤ of any 𝑛-mode state 𝜌, and poly(𝑛, 1/𝜖) time to output a pure Gaussian state | 𝜓⟩ opt + 𝜖 with probability at least 0.99. Í The key trick is to define the projector 𝑃𝐺 = {𝑖, 𝑗 } ∈𝐸 𝑎ˆ †𝑖 𝑎ˆ †𝑗 |0⟩⟨0| 𝑎ˆ 𝑗 𝑎ˆ 𝑖 which projects onto the two-photon excited states corresponding with the edges of 𝐺. Considering its expectation value with a pure Gaussian 54

state via (6), ∑︁

⟨𝜓(𝑏, 𝐴)| 𝑃𝐺 |𝜓(𝑏, 𝐴)⟩ =

|⟨𝜓(𝑏, 𝐴)| 𝑎ˆ †𝑖 𝑎ˆ †𝑗 |0⟩| 2 =

{𝑖, 𝑗 } ∈𝐸

1 𝑍 (𝑏, 𝐴)

∑︁

| 𝐴𝑖 𝑗 + 𝑏 𝑖 𝑏 𝑗 | 2

{𝑖, 𝑗 } ∈𝐸

We now observe that restricting to the family 𝑏(𝑥; 𝛽) = 𝛽𝑥 and 𝐴(𝑥; 𝛼) = 𝛼𝑥𝑥 ⊤ for 𝛽 > 0 and 𝛼 ∈ C collapses the summation to exactly the maximand of (78): ⟨𝜓(𝑥; 𝛽, 𝛼)| 𝑃𝐺 |𝜓(𝑥; 𝛽, 𝛼)⟩ =

|𝛼 + 𝛽2 | 2 𝑍 (𝛽𝑥, 𝛼𝑥𝑥 ⊤ )

∑︁

|𝑥𝑖 | 2 |𝑥 𝑗 | 2

(79)

{𝑖, 𝑗 } ∈𝐸

We now claim that 𝑍 (𝛽𝑥, 𝛼𝑥𝑥 ⊤ ) is independent of 𝑥. Applying the passive unitary which carries (𝛽𝑥, 𝛼𝑥𝑥 ⊤ ) → ′ ′ (𝛽𝑒 1 , 𝛼𝑒 1 𝑒 ⊤ 1 ) ≡ (𝑏 , 𝐴 ), 1 log 𝑍 (𝛽𝑥, 𝛼𝑥𝑥 ⊤ ) = log 𝑍 (𝑏 ′ , 𝐴′ ) = − log det 𝑀 ( 𝐴′ ) + 𝑢(𝑏 ′ ) ⊤ 𝑀 ( 𝐴′ ) −1 𝑢(𝑏 ′ ) 2   1 1 + Re 𝛼 = − log 1 − |𝛼| 2 + 𝛽2 2 1 − |𝛼| 2  2  (1+Re 𝛼) exp 𝛽 1− | 𝛼| 2 𝑍 (𝛽, 𝛼) ≡ 𝑍 (𝛽𝑥, 𝛼𝑥𝑥 ⊤ ) = √︁ 1 − |𝛼| 2

(80)

   𝐼 − Re{𝛼}𝑒 1 𝑒 ⊤ Im{𝛼}𝑒 1 𝑒 ⊤ 𝛽𝑒 1 ′ 1 1 and 𝑢(𝑏 ) = to simplify. Im{𝛼}𝑒 1 𝑒 ⊤ 𝐼 + Re{𝛼}𝑒 1 𝑒 ⊤ 0 1 1 Thus, the maximal value of ⟨𝜓(𝑥; 𝛽, 𝛼)| 𝑃𝐺 |𝜓(𝑥; 𝛽, 𝛼)⟩ for any fixed (𝛼, 𝛽) gives a known expression involving 𝜔(𝐺). Unfortunately, our agnostic learner optimizes over all pure Gaussian states as opposed to just this family, so even if we let our arbitrary state be 𝜌 = 𝑃𝐺 /tr(𝑃𝐺 ), it is possible that a pure Gaussian state outside of this family may exploit squeezing to achieve a significantly larger expectation against 𝑃𝐺 than the best possible in this family. Thus, we would like to perturb 𝜌 with a term that discourages pure Gaussian states outside of this family without biasing any particular member of the family. Since all |𝜓(𝑥; 𝛽, 𝛼)⟩ for a fixed (𝛽, 𝛼) can reach each other via a passive unitary, we expect this term to have an expectation value which is invariant under passive changes of modes. Í𝑛 † A natural choice is the one-photon projector 𝑃1 = 𝑖=1 𝑎ˆ 𝑖 |0⟩⟨0| 𝑎ˆ 𝑖 . In particular, we prove that there exist (𝛼∗ , 𝛽∗ ) such that any |𝜓⟩ ∈ G that nearly maximizes 𝑝 1 (|𝜓⟩) = ⟨𝜓| 𝑃1 |𝜓⟩ must be close to a state of the form |𝜓(𝑥; 𝛽∗ , 𝛼∗ )⟩, with all maximizers taking exactly this form. √︁ Lemma 6.3. Let (𝛽∗ , 𝛼∗ ) = ( 3/2, −1/2) and abbreviate |𝜓 ∗ (𝑥)⟩ ≡ |𝜓(𝑥; 𝛽∗ , 𝛼∗ )⟩ for ∥𝑥∥ 2 = 1. Then, for any |𝜓(𝑏, 𝐴)⟩ ∈ G with 𝑏 ≠ 0, where we use Lemma 2.6 with 𝑀 ( 𝐴′ ) =

𝑝 1 (|𝜓⟩) ≤ 𝑝 ∗1 ,



inf 𝑑IF (|𝜓⟩ , |𝜓 ∗ (𝑥)⟩) ≲ 𝑝 ∗1 − 𝑝 1 (|𝜓⟩)

∥ 𝑥 ∥ 2 =1

√ where 𝑝 ∗1 = 3 3/(4𝑒) is the maximal value achieved by states exactly of the form |𝜓(𝑥)⟩. We defer the proof to Appendix A.5.1.

55

Taking stock of our progress, the one-photon projector 𝑃1 heavily biases pure Gaussian states of the form ∗ |𝜓 (𝑥)⟩ but does not discriminate a particular 𝑥 ∈ C𝑛 . On the other hand, 𝑃𝐺 , when restricted to the family |𝜓 ∗ (𝑥)⟩, selects out the 𝑥 that maximizes the left hand side of (78), yielding an expression for 𝜔(𝐺). Thus, we take our arbitrary state 𝜌 to be 𝑃1 with a small perturbation in the direction 𝑃𝐺 : 𝜌𝐺 =

𝑃1 + 𝜆𝑃𝐺 𝑛 + 𝜆|𝐸 |

(81)

which clearly has tr 𝜌𝐺 = 1. We now use the stability of optimizing 𝑝 1 (·) and (79) to relate the optimum of our pure Gaussian state optimization problem to 𝜔(𝐺). Í Lemma 6.4. Let OPT𝐺 = max ∥ 𝑥 ∥ 2 =1 {𝑖, 𝑗 } ∈𝐸 |𝑥𝑖 | 2 |𝑥 𝑗 | 2 and OPT𝜌𝐺 ≡ max | 𝜓⟩ ∈ G 𝐹 (𝜌𝐺 , |𝜓⟩⟨𝜓|). Then, 𝑝 ∗1 +

2 ∗ 2 𝑝 1 𝜆OPT𝐺 ≤ (𝑛 + 𝜆|𝐸 |) OPT𝜌𝐺 ≤ 𝑝 ∗1 + 𝑝 ∗1 𝜆OPT𝐺 + 𝐶𝜆2 3 3

(82)

for universal constant 𝐶 > 0. Proof. Define 𝑝 𝐺 (|𝜓⟩) ≡ ⟨𝜓| 𝑃𝐺 |𝜓⟩ such that: (𝑛 + 𝜆|𝐸 |)OPT𝜌𝐺 = max ( 𝑝 1 (|𝜓⟩) + 𝜆𝑝 𝐺 (|𝜓⟩)) | 𝜓⟩ ∈ G

Then, the lower bound immediately follows from restricting to the family |𝜓 ∗ (𝑥)⟩:   ∗ ) 2 (1 + Re 𝛼 ∗ ) √︁ (𝛽 ∗ ∗ ∗ max ( 𝑝 1 (|𝜓 (𝑥)⟩) + 𝜆𝑝 𝐺 (|𝜓 (𝑥)⟩)) = 𝑝 1 + 𝜆 1 − |𝛼∗ | 2 exp − |𝛼∗ + (𝛽∗ ) 2 | 2 OPT𝐺 | 𝜓∗ ( 𝑥 ) ⟩ 1 − |𝛼∗ | 2 √ 3 ∗ = 𝑝 1 + 𝜆 OPT𝐺 2𝑒 2 (83) = 𝑝 ∗1 + 𝜆𝑝 ∗1 OPT𝐺 3 where the first term is given by Lemma 6.3 and the second follows from (79) and (80). For the upper bound, we leverage Lemma 6.3 to control the performance of |𝜓⟩ ∈ G. Specifically, suppose |𝜓⟩ fails to meet the optimum value of 𝑝 1 (·) by Δ1 = 𝑝 ∗1 − 𝑝 1 (|𝜓⟩). Then, by Lemma 6.3, there exists an 𝑥 ∈ C𝑛 with ∥𝑥∥ 2 = 1 such that 𝑑IF (|𝜓⟩ , |𝜓 ∗ (𝑥)⟩) ≲ Δ1 . Using this to control the deviation of 𝑝 𝐺 (·) from the optimum: √︁ √︁ | 𝑝 𝐺 (|𝜓 ∗ (𝑥)⟩) − 𝑝 𝐺 (|𝜓⟩)| ≤ 𝑑tr (|𝜓⟩⟨𝜓| , |𝜓 ∗ (𝑥)⟩⟨𝜓 ∗ (𝑥)|) = 𝑑IF (|𝜓⟩ , |𝜓 ∗ (𝑥)⟩) ≤ 𝐶1 Δ1 for some constant 𝐶1 > 0. Applying this, √︁ √︁ 𝐶2 2 𝑝 1 (|𝜓⟩) + 𝜆𝑝 𝐺 (|𝜓⟩) ≤ 𝑝 ∗1 − Δ + 𝜆𝑝 𝐺 (|𝜓 ∗ (𝑥)⟩) + 𝜆𝐶1 Δ1 ≤ 𝑝 ∗1 + 𝜆𝑝 ∗1 OPT𝐺 − ( Δ1 − 𝜆𝐶1 /2) 2 + 1 𝜆2 3 4 2 ≤ 𝑝 ∗1 + 𝜆𝑝 ∗1 OPT𝐺 + 𝐶1 𝜆2 3 where we bound 𝑝 𝐺 (|𝜓 ∗ (𝑥)⟩) ≤ max | 𝜓∗ ( 𝑥 ) ⟩ 𝑝 𝐺 (|𝜓 ∗ (𝑥)⟩) = 23 𝜆𝑝 ∗1 OPT𝐺 from the computation in (83), 56

complete the square on the error terms, and define 𝐶 accordingly. This proves the claim.

□

We are now ready to reduce maximum clique. Proof of Theorem 6.2. Suppose we have a polytime agnostic learner for pure Gaussian states to error opt + 𝜖. 𝜆 From (81), given 𝐺, we can prepare a copy of 𝜌 by preparing 𝑎ˆ †𝑖 𝑎ˆ †𝑗 |0⟩ with probability 𝑛+𝜆| 𝐸 | for {𝑖, 𝑗 } ∈ 𝐸 † 𝑛 and 𝑎ˆ 𝑖 |0⟩ with probability 𝑛+𝜆| 𝐸 | for 𝑖 ∈ [𝑛]. Feeding poly(𝑛, 1/𝜖1 ) copies of 𝜌𝐺 to our learner gives a 𝜓ˆ ˆ is a pure Gaussian state, we can then use Lemma 3.12 ˆ 𝜓|) ˆ ≥ OPT𝜌𝐺 − 𝜖. Since | 𝜓⟩ such that 𝐹 (𝜌𝐺 , | 𝜓⟩⟨ −2 ˜ ˆ 𝜓|) ˆ to additive error 𝜖, using Gaussian unitaries and on-off with 𝑂 (𝜖 ) copies of 𝜌𝐺 to estimate 𝐹 (𝜌𝐺 , | 𝜓⟩⟨ photon detection. It follows that we can estimate OPT𝜌𝐺 to additive error 𝜖 in poly(𝑛, 1/𝜖) time. d 𝜌𝐺 , form the following estimates: Calling this estimate OPT 3 d 𝜌𝐺 − 𝑝 ∗ ) ((𝑛 + 𝜆|𝐸 |) OPT 1 2𝑝 ∗1 𝜆   1 š = round 𝜔(𝐺) d𝐺 1 − 2OPT

d𝐺 = OPT

Viewing OPT𝐺 as a function of 𝜔(𝐺),we have that OPT𝐺 (𝜔 + 1) − OPT𝐺 (𝜔) = 2𝜔 ( 1𝜔+1) ≳ 𝑛 −2 . Since š = 𝜔(𝐺) if | OPT d 𝐺 − OPT𝐺 | ≤ 𝑐𝑛 −2 for small constant 𝑐 > 0. Re-arranging (82), 𝜔(𝐺) is an integer, 𝜔(𝐺) d 𝐺 − OPT𝐺 | ≤ | OPT

3 (𝐶𝜆2 + (𝑛 + 𝜆|𝐸 |)𝜖) ≲ 𝜆 + (𝑛/𝜆 + 𝑛2 )𝜖 2𝑝 ∗1 𝜆

Thus, taking 𝜆 = 𝑂 (𝑛 −2 ) and 𝜖 = 𝑂 (𝑛 −5 ) suffices. This gives an efficient algorithm for computing 𝜔(𝐺), which is NP-hard. □

AI Usage Disclosure Multiple technical ideas were derived in discussions with GPT-5.6 and GPT-6. In multiple conversations with GPT, the authors proposed the overall framework, the connection with robust statistics, pruning, and polynomial optimization, and the details of these ideas were then worked out in collaboration with GPT. The preparation of the manuscript was also assisted by Claude Opus 5.5 and Fable 5.1. The authors claim responsibility for any errors and omissions in the manuscript.

57

References [AA24] Anurag Anshu and Srinivasan Arunachalam. A survey on the complexity of learning quantum states. Nature Reviews Physics, 6(1):59–69, 2024. [AAA+ 19] Fausto Acernese, M. Agathos, L. Aiello, A. Allocca, A. Amato, S. Ansoldi, S. Antier, M. Arène, N. Arnaud, S. Ascenzi, et al. Increasing the astrophysical reach of the advanced Virgo detector via the application of squeezed vacuum states of light. Physical Review Letters, 123(23):231108, 2019. [AAA+ 25] H. Aghaee Rad, T. Ainsworth, R. N. Alexander, B. Altieri, et al. Scaling and networking a modular photonic quantum computer. Nature, 638:912–919, 2025. [ABCL25] Maryam Aliakbarpour, Vladimir Braverman, Nai-Hui Chia, and Yuhan Liu. Adversarially robust quantum state learning and testing. arXiv preprint arXiv:2508.13959, 2025. [ADGG26] Srinivasan Arunachalam, Arkopal Dutt, Sabee Grewal, and Aparna Gupte. Marton’s conjecture in polynomial time. arXiv preprint arXiv:2609.20771, 2026. [ADKL25] Alvan Arulandu, Ilias Diakonikolas, Daniel Kane, and Jerry Li. Agnostic product mixed state tomography via robust statistics. arXiv preprint arXiv:2510.08472, 2025. [AGKE15] Leandro Aolita, Christian Gogolin, Martin Kliesch, and Jens Eisert. Reliable quantum certification of photonic state preparations. Nature Communications, 6:8498, 2015. [AGPF18] Francesco Albarelli, Marco G. Genoni, Matteo G. A. Paris, and Alessandro Ferraro. Resource theory of quantum non-Gaussianity and Wigner negativity. Physical Review A, 98(5):052350, 2018. [BBK+ 25] Ainesh Bakshi, John Bostanci, William Kretschmer, Zeph Landau, Jerry Li, Allen Liu, Ryan O’Donnell, and Ewin Tang. Learning the closest product state. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 1212–1221, 2025. [BBV08] Maria-Florina Balcan, Avrim Blum, and Santosh Vempala. Clustering via similarity functions: theoretical foundations and algorithms. In Proceedings of the 40 th ACM Symposium on Theory of Computing (STOC), 2008. [BHN26] Zongbo Bao, Jonas Helsen, and Tuyen Nguyen. Fully tolerant product state testing and closest product state learning. arXiv preprint arXiv:2610.01979, 2026. [BMEL25] Lennart Bittel, Antonio Anna Mele, Jens Eisert, and Lorenzo Leone. Optimal tracedistance bounds for free-fermionic states: Testing and improved tomography. PRX Quantum, 6(3):030341, 2025. [BMEM25] Lennart Bittel, Francesco A Mele, Jens Eisert, and Antonio A Mele. Energy-independent tomography of gaussian states. arXiv preprint arXiv:2508.14979, 2025. [BO21] Costin Bădescu and Ryan O’Donnell. Improved quantum data analysis. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 1398–1411, 2021. 58

[Bog47] N. N. Bogoliubov. On the theory of superfluidity. Journal of Physics (USSR), 11:23, 1947. [BPK+ 21] K. M. Backes, D. A. Palken, S. Al Kenany, B. M. Brubaker, S. B. Cahn, A. Droster, Gene C. Hilton, Sumita Ghosh, H. Jackson, S. K. Lamoreaux, et al. A quantum enhanced search for dark matter axions. Nature, 590(7845):238–242, 2021. [Bub15] Sébastien Bubeck. Convex optimization: Algorithms and complexity. Foundations and Trends in Machine Learning, 8(3–4):231–357, 2015. [CFG+ 26] Senrui Chen, Marco Fanizza, Filippo Girardi, Ludovico Lami, Francesco Anna Mele, Michael Walter, and Freek Witteveen. Optimal tomography of bosonic and fermionic gaussian states. arXiv preprint arXiv:2607.11847, 2026. [CGYZ25] Sitan Chen, Weiyuan Gong, Qi Ye, and Zhihan Zhang. Stabilizer bootstrapping: A recipe for efficient agnostic tomography and magic estimation. In Proceedings of the 57th Annual ACM Symposium on Theory of Computing, pages 429–438, 2025. [CGYZ26] Sitan Chen, Weiyuan Gong, Qi Ye, and Zhihan Zhang. The log log jam in gaussian state tomography. arXiv preprint arXiv:2607.12983, 2026. [CIET+ 20] P. Campagne-Ibarcq, A. Eickbusch, S. Touzard, E. Zalys-Geller, N. E. Frattini, V. V. Sivak, P. Reinhold, S. Puri, S. Shankar, R. J. Schoelkopf, L. Frunzio, M. Mirrahimi, et al. Quantum error correction of a qubit encoded in grid states of an oscillator. Nature, 584(7821):368–372, 2020. [CMF+ 26] Senrui Chen, Francesco Anna Mele, Marco Fanizza, Alfred Li, Zachary Mann, Hsin-Yuan Huang, Yanbei Chen, and John Preskill. Towards sample-optimal learning of bosonic gaussian quantum states. arXiv preprint arXiv:2603.18136, 2026. [CMG07] Robert L. Cook, Paul J. Martin, and J. M. Geremia. Optical coherent state discrimination using a closed-loop quantum measurement. Nature, 446(7137):774–777, 2007. [CMG20a] Ulysse Chabaud, Damian Markham, and Frédéric Grosshans. Stellar representation of nonGaussian quantum states. Physical Review Letters, 124(6):063605, 2020. [CMG20b] Ulysse Chabaud, Damian Markham, and Frédéric Grosshans. Stellar representation of nonGaussian quantum states. Physical Review Letters, 124:063605, 2020. Supplemental section and equation numbers refer to arXiv:1907.11009v3. [CMMP26] Senrui Chen, Antonio Anna Mele, Francesco Anna Mele, and John Preskill. When are bosonic Gaussian states classical to learn? arXiv preprint arXiv:2609.26705, 2026. [CRW+ 21] Ulysse Chabaud, Ganaël Roeland, Mattia Walschaers, Frédéric Grosshans, Valentina Parigi, Damian Markham, and Nicolas Treps. Certification of non-Gaussian states with operational measurements. PRX Quantum, 2(2):020333, 2021. [CS96] Gregory Constantine and Thomas Savits. A multivariate faa di bruno formula with applications. Transactions of the American Mathematical Society, 348(2):503–520, 1996. 59

[CSV17] Moses Charikar, Jacob Steinhardt, and Gregory Valiant. Learning from untrusted data. In Proceedings of the 49th annual ACM SIGACT symposium on theory of computing, pages 47– 60, 2017. [DBC+ 13] S. Dwyer, L. Barsotti, S. S. Y. Chua, M. Evans, et al. Squeezed quadrature fluctuations in a gravitational wave detector using squeezed light. Optics Express, 21(16):19047, 2013. [DK19] Ilias Diakonikolas and Daniel M Kane. Recent advances in algorithmic high-dimensional robust statistics. arXiv preprint arXiv:1911.05911, 2019. [DK23] Ilias Diakonikolas and Daniel M Kane. Algorithmic high-dimensional robust statistics. Cambridge University Press, 2023. [DKK+ 16] Ilias Diakonikolas, Gautam Kamath, Daniel Kane, Jerry Li, Ankur Moitra, and Alistair Stewart. Robust estimators in high-dimensions without the computational intractability. FOCS 2016, SIAM Journal on Computing, 2016. [DKP20] Ilias Diakonikolas, Daniel M Kane, and Ankit Pensia. Outlier robust mean estimation with subgaussian rates via stability. Advances in neural information processing systems, 33:1830– 1840, 2020. [DKS17] Ilias Diakonikolas, Daniel M Kane, and Alistair Stewart. Statistical query lower bounds for robust estimation of high-dimensional gaussians and gaussian mixtures. In 2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS), pages 73–84. IEEE, 2017. [DKS18] Ilias Diakonikolas, Daniel M. Kane, and Alistair Stewart. List-decodable robust mean estimation and learning mixtures of spherical gaussians. In Ilias Diakonikolas, David Kempe, and Monika Henzinger, editors, Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 1047–1060. ACM, 2018. [EP03] Jens Eisert and MB1069 Plenio. Introduction to the basics of entanglement theory in continuousvariable systems. International Journal of Quantum Information, 1(04):479–506, 2003. [FHD+ 06] Alexander Franzen, Boris Hage, James DiGuglielmo, Jaromı́r Fiurášek, and Roman Schnabel. Experimental demonstration of continuous variable purification of squeezed states. Physical Review Letters, 97(15):150505, 2006. [FL18] Shmuel Friedland and Lek-Heng Lim. Nuclear norm of higher-order tensors. Mathematics of Computation, 87(311):1255–1281, 2018. [FM11] Radim Filip and Ladislav Mišta. Detecting quantum states with a positive Wigner function beyond mixtures of Gaussian states. Physical Review Letters, 106(20):200401, 2011. [FNB+ 24] Jiřı́ Fadrný, Michal Neset, Martin Bielak, Miroslav Ježek, Jan Bı́lek, and Jaromı́r Fiurášek. Experimental preparation of multiphoton-added coherent states of light. npj Quantum Information, 10(1):89, 2024. [FNM+ 19] C. Flühmann, T. L. Nguyen, M. Marinelli, V. Negnevitsky, K. Mehta, and J. P. Home. Encoding a qubit in a trapped-ion mechanical oscillator. Nature, 566(7745):513–517, 2019. 60

[GDD+ 13] Hartmut Grote, Karsten Danzmann, Kathrine L. Dooley, Roman Schnabel, Jacob Slutsky, and Henning Vahlbruch. First long-term application of squeezed states of light in a gravitationalwave observatory. Physical Review Letters, 110(18):181101, 2013. [GFP+ 20] A. Grimm, N. E. Frattini, S. Puri, S. O. Mundhada, S. Touzard, M. Mirrahimi, S. M. Girvin, S. Shankar, and M. H. Devoret. Stabilization and operation of a Kerr-cat qubit. Nature, 584(7820):205–209, 2020. [GIKL24a] Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang. Agnostic tomography of stabilizer product states. arXiv preprint arXiv:2404.03813, 2024. [GIKL24b] Sabee Grewal, Vishnu Iyer, William Kretschmer, and Daniel Liang. Improved stabilizer estimation via Bell difference sampling. In Proceedings of the 56th Annual ACM Symposium on Theory of Computing, pages 1352–1363, 2024. [GJN+ 23] D. Ganapathy, W. Jia, M. Nakano, V. Xu, et al. Broadband quantum enhancement of the LIGO detectors with frequency-dependent squeezing. Physical Review X, 13:041021, 2023. [GKP01] Daniel Gottesman, Alexei Kitaev, and John Preskill. Encoding a qubit in an oscillator. Physical Review A, 64(1):012310, 2001. [GMM13] Iulia Ghiu, Paulina Marian, and Tudor A. Marian. Measures of non-Gaussianity for one-mode field states. Physica Scripta, T153:014028, 2013. [GPB07] Marco G. Genoni, Matteo G. A. Paris, and Konrad Banaszek. Measure of the non-Gaussian character of a quantum state. Physical Review A, 76(4):042327, 2007. [GPB08] Marco G. Genoni, Matteo G. A. Paris, and Konrad Banaszek. Quantifying the non-Gaussian character of a quantum state by quantum relative entropy. Physical Review A, 78(6):060303, 2008. [GWM+ 25] Filippo Girardi, Freek Witteveen, Francesco Anna Mele, Lennart Bittel, Salvatore F. E. Oliviero, David Gross, and Michael Walter. Is it Gaussian? testing bosonic quantum states. arXiv preprint arXiv:2510.07305, 2025. [HGS+ 20] Lucas Hackl, Tommaso Guaita, Tao Shi, Jutho Haegeman, Eugene Demler, and Ignacio Cirac. Geometry of variational methods: dynamics of closed quantum systems. SciPost Physics, 9(4):048, 2020. [HKS+ 17] Craig S. Hamilton, Regina Kruse, Linda Sansoni, Sonja Barkhofen, Christine Silberhorn, and Igor Jex. Gaussian boson sampling. Physical Review Letters, 119(17):170501, 2017. [HT25] Oliver Hahn and Ryuji Takagi. Measuring non-Gaussianity with correlation. arXiv preprint arXiv:2508.19890, 2025. [Hub64] Peter J Huber. Robust estimation of a location parameter. The Annals of Mathematical Statistics, 35(1):73–101, 1964.

61

[IK22] Misha Ivkov and Pravesh K Kothari. List-decodable covariance estimation. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 1276–1283, 2022. [IOP+ 26] Vishnu Iyer, Bryan O’Gorman, Ojas Parekh, Kevin Thompson, and Andrew Zhao. Optimal testing of fermionic and bosonic Gaussian states. arXiv preprint arXiv:2609.39113, 2026. [JSM+ 11] Miroslav Ježek, Ivo Straka, Michal Mičuda, Miloslav Dušek, Jaromı́r Fiurášek, and Radim Filip. Experimental test of the quantum non-Gaussian character of a heralded single-photon state. Physical Review Letters, 107(21):213602, 2011. [KAH+ 24] Shunya Konno, Warit Asavanant, Fumiya Hanamura, Hironari Nagayoshi, Kosuke Fukui, Atsushi Sakaguchi, Ryuhoh Ide, Fumihiro China, Masahiro Yabuno, Shigehito Miki, Hirotaka Terai, Kan Takase, et al. Logical states for fault-tolerant quantum computation with propagating light. Science, 383(6680):289–293, 2024. [KMI+ 14] P. Kwee, J. Miller, T. Isogai, L. Barsotti, and M. Evans. Decoherence and degradation of squeezed states in quantum filter cavities. Physical Review D, 90(6):062006, 2014. [KMZ22] Pravesh K Kothari, Peter Manohar, and Brian Hu Zhang. Polynomial-time sum-of-squares can robustly estimate mean and covariance of gaussians optimally. In International Conference on Algorithmic Learning Theory, pages 638–667. PMLR, 2022. [KVL+ 13] Gerhard Kirchmair, Brian Vlastakis, Zaki Leghtas, Simon E. Nigg, Hanhee Paik, Eran Ginossar, Mazyar Mirrahimi, Luigi Frunzio, S. M. Girvin, and R. J. Schoelkopf. Observation of quantum state collapse and revival due to the single-photon Kerr effect. Nature, 495(7440):205–209, 2013. [Lam21] Ludovico Lami. Quantum data hiding with continuous-variable systems. Physical Review A, 104(5):052428, 2021. [LBK+ 25] M. V. Larsen, J. E. Bourassa, S. Kocsis, J. F. Tasker, et al. Integrated photonic source of Gottesman–Kitaev–Preskill qubits. Nature, 642:587–591, 2025. [LGO+ 20] A. I. Lvovsky, Philippe Grangier, Alexei Ourjoumtsev, Valentina Parigi, Masahide Sasaki, and Rosa Tualle-Brouri. Production and applications of non-Gaussian quantum states of light. arXiv preprint arXiv:2006.16985, 2020. [LRV16] Kevin A. Lai, Anup B. Rao, and Santosh Vempala. Agnostic estimation of mean and covariance. In Proceedings of FOCS’16, 2016. [LSH+ 19] Lukáš Lachman, Ivo Straka, Josef Hloušek, Miroslav Ježek, and Radim Filip. Faithful hierarchy of genuine 𝑛-photon quantum non-Gaussian light. Physical Review Letters, 123(4):043601, 2019. [LVP+ 20] Raphaël Lescanne, Marius Villiers, Théau Peronnin, Alain Sarlette, Matthieu Delbecq, Benjamin Huard, Takis Kontos, Mazyar Mirrahimi, and Zaki Leghtas. Exponential suppression of bit-flips in a qubit encoded in an oscillator. Nature Physics, 16(5):509–513, 2020.

62

[MBGF25] Francesco Anna Mele, Giovanni Barbarino, Vittorio Giovannetti, and Marco Fanizza. Achievable rates in non-asymptotic bosonic quantum communication. arXiv preprint arXiv:2502.05524, 2025. [Mel26] Francesco Anna Mele. Advances in quantum learning theory with bosonic systems. arXiv preprint arXiv:2605.08082, 2026. [MLA+ 22] Lars S. Madsen, Fabian Laudenbach, Mohsen Falamarzi Askarani, Fabien Rortais, Trevor Vincent, Jacob F. F. Bulmer, Filippo M. Miatto, Leonhard Neuhaus, Lukas G. Helt, Matthew J. Collins, et al. Quantum computational advantage with a programmable photonic processor. Nature, 606(7912):75–81, 2022. [MMB+ 25] Francesco A Mele, Antonio A Mele, Lennart Bittel, Jens Eisert, Vittorio Giovannetti, Ludovico Lami, Lorenzo Leone, and Salvatore FE Oliviero. Learning quantum states of continuousvariable systems. Nature Physics, pages 1–7, 2025. [MOUC25] Francesco Anna Mele, Salvatore Francesco Emanuele Oliviero, Varun Upreti, and Ulysse Chabaud. The symplectic rank of non-Gaussian quantum states. arXiv preprint arXiv:2504.19319, 2025. [MPB+ 19] M. Malnou, D. A. Palken, B. M. Brubaker, Leila R. Vale, Gene C. Hilton, and K. W. Lehnert. Squeezed vacuum used to accelerate the search for a weak classical signal. Physical Review X, 9(2):021023, 2019. [MS65] Theodore S Motzkin and Ernst G Straus. Maxima for graphs and a new proof of a theorem of turán. Canadian Journal of Mathematics, 17:533–540, 1965. [PNH+ 25] Harald Putterman, Kyungjoo Noh, Connor T. Hann, Gregory S. MacCabe, et al. Hardwareefficient quantum error correction via concatenated bosonic qubits. Nature, 638:927–934, 2025. [PP+ 08] Kaare Brandt Petersen, Michael Syskind Pedersen, et al. The matrix cookbook. Technical University of Denmark, 7(15):510, 2008. [PZD26] Nisarga Paul, Haimeng Zhao, and David D Dai. Learning the closest slater determinant. arXiv preprint arXiv:2607.20623, 2026. [SDC18] Tao Shi, Eugene Demler, and J. Ignacio Cirac. Variational study of fermionic and bosonic systems with non-Gaussian states: Theory and applications. Annals of Physics, 390:245–302, 2018. [Ser23a] Alessio Serafini. Quantum continuous variables: a primer of theoretical methods. CRC press, 2023. [SER+ 23b] V. V. Sivak, A. Eickbusch, B. Royer, S. Singh, I. Tsioutsios, S. Ganjam, A. Miano, B. L. Brock, A. Z. Ding, L. Frunzio, S. M. Girvin, R. J. Schoelkopf, et al. Real-time quantum error correction beyond break-even. Nature, 616(7955):50–55, 2023.

63

[SW21] Krishna Kumar Sabapathy and Andreas Winter. Bosonic data hiding: Power of linear vs non-linear optics. arXiv preprint arXiv:2102.01622, 2021. [Tim14] Aleksandr Filippovich Timan. Theory of approximation of functions of a real variable, volume 34. Elsevier, 2014. [TS08] Masahiro Takeoka and Masahide Sasaki. Discrimination of the binary coherent signal: Gaussian-operation limit and simple non-Gaussian near-optimal receivers. Physical Review A, 78(2):022320, 2008. [Tuk60] John Wilder Tukey. A survey of sampling from contaminated distributions. Contributions to probability and statistics, pages 448–485, 1960. [TYK+ 19] M. Tse, Haocun Yu, N. Kijbunchoo, A. Fernandez-Galiana, P. Dupej, L. Barsotti, C. D. Blair, D. D. Brown, S. E. Dwyer, A. Effler, et al. Quantum-enhanced advanced LIGO detectors in the era of gravitational-wave astronomy. Physical Review Letters, 123(23):231107, 2019. [TZ18] Ryuji Takagi and Quntao Zhuang. Convex resource theory of non-Gaussianity. Physical Review A, 97(6):062337, 2018. [VKL+ 13] Brian Vlastakis, Gerhard Kirchmair, Zaki Leghtas, Simon E. Nigg, Luigi Frunzio, S. M. Girvin, Mazyar Mirrahimi, M. H. Devoret, and R. J. Schoelkopf. Deterministically encoding quantum information using 100-photon Schrödinger cat states. Science, 342(6158):607–610, 2013. [Wal21] Mattia Walschaers. Non-Gaussian quantum states and where to find them. PRX Quantum, 2(3):030204, 2021. [WLKD25] Chirag Wadhwa, Laura Lewis, Elham Kashefi, and Mina Doosti. Agnostic process tomography. PRX Quantum, 6(4):040371, 2025. [WY19] Yihong Wu and Pengkun Yang. Chebyshev polynomials, moment matching, and optimal estimation of the unseen. The Annals of Statistics, 47(2):857–883, 2019. [YPH+ 26] Mahtab Yaghubi Rad, Ricard Puig, Saksham Hassanandani, Luke Coffman, Jens Eisert, Carlos Bravo-Prieto, and Antonio Anna Mele. Testing quantum Gaussianity with constant sample complexity. arXiv preprint arXiv:2609.40270, 2026. [ZWD+ 20] Han-Sen Zhong, Hui Wang, Yu-Hao Deng, Ming-Cheng Chen, Li-Chao Peng, Yi-Han Luo, Jian Qin, Dian Wu, Xing Ding, Yi Hu, et al. Quantum computational advantage using photons. Science, 370(6523):1460–1463, 2020.

64

A

Deferred proofs

A.1

Deferred proofs from Section 2

A.1.1

Proof of Lemma 2.5

Proof. In the Bargmann representation, 𝑎ˆ 𝑗 acts as 𝜕𝑧 𝑗 and 𝑎ˆ †𝑘 acts as multiplication by 𝑧 𝑘 . Differentiating the exponential in (5), using the symmetry of 𝐴, gives ! ∑︁ ⊤ ⊤ 𝑏⊤ 𝑧+𝑧 ⊤ 𝐴𝑧/2 𝜕𝑧 𝑗 𝑒 = 𝑏𝑗 + 𝐴 𝑗 𝑘 𝑧 𝑘 𝑒 𝑏 𝑧+𝑧 𝐴𝑧/2 . 𝑘

Consequently, with 𝑎ˆ = ( 𝑎ˆ 1 , . . . , 𝑎ˆ 𝑛 ) ⊤ and 𝑎ˆ † denoting the column of creation operators, (𝑎 − 𝐴𝑎 † ) |𝜓⟩ = 𝑏 |𝜓⟩ .

(84)

¯ so substitution ¯ Its conjugate gives 𝛽¯ = 𝑏¯ + 𝐴𝛽, Hitting this whole expression by ⟨𝜓| yields 𝑏 = 𝛽 − 𝐴 𝛽. † ¯ ¯ ¯ ¯ yields (𝐼 − 𝐴 𝐴) 𝛽 = 𝑏 + 𝐴 𝑏. Since 𝐴 is symmetric, 𝐴 = 𝐴 and 𝐼 − 𝐴 𝐴 is positive definite, proving the inverse formula in (7). The triangle and reverse triangle inequalities applied to 𝛽 − 𝐴 𝛽¯ prove (8). We next prove the photon-number formula (9). By definition, 𝜈(𝜋 𝑏, 𝐴) is the expectation of the total number operator in the normalized state |𝜓⟩: 𝑛 𝑛 𝑛 ∑︁ ∑︁ ∑︁ © † ª † ⟨𝜓| 𝑎ˆ 𝑗 𝑎ˆ 𝑗 |𝜓⟩ = 𝜈(𝜋 𝑏, 𝐴) = tr ­𝜋 𝑏, 𝐴 𝑎ˆ 𝑗 𝑎ˆ 𝑗 ® = ∥ 𝑎ˆ 𝑗 |𝜓⟩ ∥ 2 . 𝑗=1 𝑗=1 « ¬ 𝑗=1

(85)

To evaluate this sum, center the annihilation operators by setting 𝑑 𝑗 = 𝑎ˆ 𝑗 − 𝛽 𝑗 𝐼. They satisfy ⟨𝑑 𝑗 ⟩ 𝜓 = 0 and [𝑑 𝑗 , 𝑑 †𝑘 ] = 𝛿 𝑗 𝑘 𝐼. Now expand 𝑎ˆ 𝑗 = 𝑑 𝑗 + 𝛽 𝑗 𝐼: ⟨𝜓| 𝑎ˆ †𝑗 𝑎ˆ 𝑗 |𝜓⟩ = ⟨𝜓| 𝑑 †𝑗 𝑑 𝑗 |𝜓⟩ + 𝛽 𝑗 ⟨𝜓| 𝑑 †𝑗 |𝜓⟩ + 𝛽¯ 𝑗 ⟨𝜓| 𝑑 𝑗 |𝜓⟩ + |𝛽 𝑗 | 2 = ∥𝑑 𝑗 |𝜓⟩ ∥ 2 + |𝛽 𝑗 | 2 . Thus (85) becomes 𝜈(𝜋 𝑏, 𝐴) = ∥ 𝛽∥ 22 +

𝑛 ∑︁

∥𝑑 𝑗 |𝜓⟩ ∥ 2 .

(86)

𝑗=1

It remains to show that

Í𝑛

𝑗=1 ∥𝑑 𝑗 |𝜓⟩ ∥

2 = Í𝑛 𝑠 2 /(1 − 𝑠 2 ). Substituting 𝑎ˆ = 𝑑 + 𝛽 𝐼 into (84) yields 𝑗 𝑗 𝑗 𝑗=1 𝑗 𝑗

(𝑑 − 𝐴𝑑 † ) |𝜓⟩ = 0. Now, choose a Takagi factorization 𝐴 = 𝑈𝑆𝑈 ⊤ , where 𝑈 is unitary and 𝑆 = diag(𝑠1 , . . . , 𝑠 𝑛 ). Define 𝑐 = 𝑈 † 𝑑, so the column of adjoints is 𝑐† = 𝑈 ⊤ 𝑑 † . Multiplication by 𝑈 † therefore gives, for each 𝑗, 𝑐 𝑗 |𝜓⟩ = 𝑠 𝑗 𝑐†𝑗 |𝜓⟩ ,

[𝑐 𝑗 , 𝑐†𝑘 ] = 𝛿 𝑗 𝑘 𝐼.

65

Letting 𝑢 𝑗 = ∥𝑐 𝑗 |𝜓⟩ ∥ 2 , we find that 𝑢 𝑗 = 𝑠2𝑗 ∥𝑐†𝑗 |𝜓⟩ ∥ 2 = 𝑠2𝑗 ⟨𝜓| 𝑐 𝑗 𝑐†𝑗 |𝜓⟩ = 𝑠2𝑗 (𝑢 𝑗 + 1),

𝑢𝑗 =

𝑠2𝑗 1 − 𝑠2𝑗

,

where the third equality follows from the fact that [𝑐 𝑗 , 𝑐†𝑗 ] = 𝐼. Since 𝑐 = 𝑈 † 𝑑 and 𝑈 is unitary, it preserves the norm of the vector (𝑑1 |𝜓⟩ , . . . , 𝑑 𝑛 |𝜓⟩) in F𝑛⊕𝑛 . Thus ∑︁

∥𝑑 𝑗 |𝜓⟩ ∥ 2 =

∑︁

∥𝑐 𝑗 |𝜓⟩ ∥ 2 =

𝑢𝑗 =

𝑗

𝑗

𝑗

∑︁

∑︁

𝑠2𝑗

𝑗

1 − 𝑠2𝑗

.

Substituting this into (86) proves (9). A.1.2

□

Proof of Lemma 2.6

Proof. Spectrum of 𝑀 ( 𝐴). Write 𝐴 = 𝑃 + 𝑖𝑄 and take Takagi vectors 𝑢 𝑗 = 𝑝 𝑗 + 𝑖𝑞 𝑗 as in (2). Taking real and imaginary parts of 𝐴𝑢¯ 𝑗 = 𝑠 𝑗 𝑢 𝑗 gives 𝑃𝑝 𝑗 + 𝑄𝑞 𝑗 = 𝑠 𝑗 𝑝 𝑗 , Direct multiplication by 𝑀 ( 𝐴) =  𝑀 ( 𝐴)



𝐼−𝑃 𝑄 𝑄 𝐼+𝑃



𝑄 𝑝 𝑗 − 𝑃𝑞 𝑗 = 𝑠 𝑗 𝑞 𝑗 .

now shows

   𝑝𝑗 𝑝𝑗 = (1 − 𝑠 𝑗 ) , −𝑞 𝑗 −𝑞 𝑗

 𝑀 ( 𝐴)

   𝑞𝑗 𝑞 = (1 + 𝑠 𝑗 ) 𝑗 . 𝑝𝑗 𝑝𝑗

These 2𝑛 real vectors form an orthonormal basis: the real and imaginary parts of 𝑢 †𝑗 𝑢 𝑘 = 𝛿 𝑗 𝑘 give 𝑝 ⊤𝑗 𝑝 𝑘 + 𝑞 ⊤𝑗 𝑞 𝑘 = 𝛿 𝑗 𝑘 ,

𝑝 ⊤𝑗 𝑞 𝑘 − 𝑞 ⊤𝑗 𝑝 𝑘 = 0.

Thus we have exhibited all the eigenvalues of 𝑀 ( 𝐴), namely 1 − 𝑠 𝑗 and 1 + 𝑠 𝑗 . They are positive because 𝑠 𝑗 ≤ ∥ 𝐴∥ op < 1. Normalization formula. To express (10) as a real integral, write 𝑧 = 𝑥 + 𝑖𝑦 and 𝑏 = 𝑏 𝑅 + 𝑖𝑏 𝐼 , keeping 𝐴 = 𝑃 + 𝑖𝑄 as above, with all components real. Since 𝐴 is symmetric, so are 𝑃 and 𝑄, and expansion gives Re(𝑧 ⊤ 𝐴𝑧) = 𝑥 ⊤ 𝑃𝑥 − 𝑦 ⊤ 𝑃𝑦 − 2𝑥 ⊤ 𝑄𝑦,

Re(𝑏 ⊤ 𝑧) = 𝑏 ⊤𝑅 𝑥 − 𝑏 ⊤𝐼 𝑦.

Consequently, for 𝑤 = (𝑥 ⊤ , 𝑦 ⊤ ) ⊤ , the exponent in (10) is − 𝑥 ⊤ (𝐼 − 𝑃)𝑥 − 𝑦 ⊤ (𝐼 + 𝑃)𝑦 − 2𝑥 ⊤ 𝑄𝑦 + 2𝑏 ⊤𝑅 𝑥 − 2𝑏 ⊤𝐼 𝑦 = −𝑤 ⊤ 𝑀 ( 𝐴)𝑤 + 2𝑢(𝑏) ⊤ 𝑤. Here 𝑑 2𝑛 𝑧 = 𝑑𝑥 𝑑𝑦 = 𝑑𝑤: we are simply identifying C𝑛 with R2𝑛 . Hence ∫ ⊤ ⊤ −𝑛 𝑍 (𝑏, 𝐴) = 𝜋 𝑒 −𝑤 𝑀 ( 𝐴) 𝑤+2𝑢(𝑏) 𝑤 𝑑𝑤. R2𝑛

66

Abbreviating 𝑀 = 𝑀 ( 𝐴) and 𝑢 = 𝑢(𝑏), complete the square: −𝑤 ⊤ 𝑀𝑤 + 2𝑢 ⊤ 𝑤 = −(𝑤 − 𝑀 −1 𝑢) ⊤ 𝑀 (𝑤 − 𝑀 −1 𝑢) + 𝑢 ⊤ 𝑀 −1 𝑢. Translating by 𝑣 = 𝑤 − 𝑀 −1 𝑢 and using the standard Gaussian integral for 𝑀 ≻ 0 gives ⊤

𝑍 (𝑏, 𝐴) = 𝜋 −𝑛 𝑒 𝑢 𝑀

−1 𝑢

∫

⊤

−1

⊤ 𝑒𝑢 𝑀 𝑢 . 𝑒 −𝑣 𝑀𝑣 𝑑𝑣 = √ R2𝑛 det 𝑀

Taking logarithms proves (11).

A.2

Deferred proofs from Section 3

A.2.1

Proof of Theorem 3.2

□

Proof. We perform a heterodyne measurement on each copy to obtain samples from a distribution 𝑃 that is opttr -close in total variation distance to the classical Gaussian N (𝜇 ℎ , Σℎ ) where 𝜇 ℎ = m and Σℎ = 𝑉+𝐼 2 . Taking opttr = optTV , we apply Proposition 2.9 to obtain estimates 𝜇ˆ ℎ and Σ̂ℎ which satisfy the relative ℓ2 2Σ̂ℎ and Frobenius guarantees in (13). Then, we construct the estimator 𝑉ˆ ≡ 1−𝜁 − 𝐼. Now, from the relative Frobenius guarantee in (13) for 𝜇ˆ ℎ and Σ̂ℎ , we have that (1−𝜁)Σℎ ⪯ Σ̂ℎ ⪯ (1+𝜁)Σℎ . Applying this, 4𝜁 16 0 ⪯ 𝑉ˆ − 𝑉 ⪯ Σℎ ⪯ 𝜁 Σℎ (87) 1−𝜁 3 ˆ is a valid where the last inequality follows from 𝜁 ≤ 1/4. This means that 𝑉ˆ + 𝑖Ω ⪰ 𝑉 + 𝑖Ω ⪰ 0, so 𝜋( 𝜇ˆ ℎ , 𝑉) Gaussian state. Then, by Fact 2.1,   𝑑tr (𝜋, 𝜋) ˆ ≲ ∥𝑉 −1/2 ( 𝜇ˆ ℎ − 𝜇 ℎ ) ∥ 2 + tr (𝑉 −1 + 𝑉ˆ −1 )|𝑉ˆ − 𝑉 |   ≤ ∥𝑉 −1/2 Σℎ1/2 ∥ op ∥Σℎ−1/2 ( 𝜇ˆ ℎ − 𝜇 ℎ ) ∥ 2 + 2 tr 𝑉 −1 (𝑉ˆ − 𝑉)   ≲ 𝜁 ∥𝑉 −1/2 Σℎ1/2 ∥ op + 𝜁 tr 𝑉 −1 Σℎ ≲ 𝜁 (𝑛 + tr 𝑉 −1 ) where the second term in the second inequality 𝑉ˆ −1 ⪯ 𝑉 −1 and where the fourth inequality uses √︃comes from

the bound ∥𝑉 −1/2 Σℎ1/2 ∥ op ≤ ∥𝑉 −1/2 Σℎ1/2 ∥ 𝐹 = tr 𝑉 −1 Σℎ and the fact that tr 𝑉 −1 Σℎ = 𝑛 + 21 tr 𝑉 −1 . Then, (16) follows by triangle inequality: 𝑑tr (𝜌, 𝜋) ˆ ≤ 𝑑tr (𝜌, 𝜋) + 𝑑tr (𝜋, 𝜋) ˆ ≲ opttr + 𝜁 (𝑛 + tr 𝑉 −1 ) ≲ (𝑛 + tr 𝑉 −1 ) · 𝑂˜ (opttr ) ≲ 𝐸 (𝜋) · 𝑂˜ (opttr )
where the final inequality comes from tr 𝑉 −1 = tr Ω𝑉 −1 Ω⊤ ≤ tr 𝑉 ≤ 4𝐸 (𝜋), with the second inequality coming from taking the Schur complement of the top right block of 𝑉 + 𝑖Ω ⪰ 0. □

67

A.2.2

Proof of Theorem 3.4

Proof. Lemmas 12 and 13 in [BMEM25] prove the following unsqueezing property which substantiates the whole algorithm in Theorem 3.3: √︄ √︂ √︁ 𝐾 5 2 −1 2𝜁 −1 −1 ∥𝑉𝑘+1 ∥ op ≤ 1 + (∥𝑉𝑘 ∥ ∞ + 1) ≤ + ∥𝑉 ∥ ∞ ⇒ ∥𝑉𝐾−1 ∥ op ≤ 7/3 · ∥𝑉 −1 ∥ 1/2 op 1−𝜁 3 3 Observing the proofs, we notice that the two sufficient claims for our purposes are showing that both 2𝜁 0 ⪯ 𝑉ˆ 𝑘 − 𝑉𝑘 ⪯ 1−𝜁 (𝑉𝑘 + 𝐼) and 𝜁 ≤ 1/4 hold for our robust estimators. Luckily, this is immediate. First, unitary invariance preserves the trace-distance bound 𝑑tr (𝜌 𝑘 , 𝜋 𝑘 ) ≤ opttr ≤ opttr = optTV across iterations, meaning these two conditions directly follow from (87) and Proposition 2.9. The claimed copy complexity then comes from the 𝑂 (𝑛2 opttr −2 log5 1/𝛿) copies needed by each usage of robust heterodyne tomography. □ A.2.3

Proof of Lemma 3.7

Proof. We begin by converting the absolute error guarantees of (19) for the heterodyne outcome distribution into the following relative guarantees on the raw, uninflated parameter estimates for the state: √ ∥𝑉 −1/2 ( m̂ − m) ∥ 2 ≤ 2𝜁 and ∥𝑉 −1/2 (𝑉ˆraw − 𝑉)𝑉 −1/2 ∥ 𝐹 ≤ 4𝜁 (88) To show this, we first recall that the eigenvalues of a pure Gaussian state’s covariance 𝑉 = 𝑆𝑆 ⊤ come in pairs of the form (𝜆, 1/𝜆), meaning ∥𝑉 −1 ∥ op ≤ 2 implies 𝐼/2 ⪯ 𝑉 ⪯ 2𝐼 and 0 ⪯ Σℎ ⪯ 3𝑉/2. Then, by standard properties of ∥ · ∥ op , √ ∥𝑉 −1/2 ( m̂ − m) ∥ 2 ≤ 2𝜁 ∥𝑉 −1/2 (𝑉ˆraw − 𝑉)𝑉 −1/2 ∥ 𝐹 ≤ 2∥𝑉ˆraw − 𝑉 ∥ 𝐹 = 4∥ Σ̂ℎ − Σℎ ∥ 𝐹 ≤ 4𝜁 proving (88). This gives 𝑉ˆraw ⪰ (1/2 − 2𝜁)𝐼 ≻ 0 for 𝜁 < 1/4 and ∥𝑉ˆraw − 𝑉 ∥ 𝐹 ≤ 2𝜁. We now prove the key inequality, ∥𝑉ˆ − 𝑉ˆraw ∥ 𝐹 ≲ 𝜁, which argues that removing the diagonal matrix in ˆ 𝑉raw = 𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ perturbs the estimate by 𝑂 (𝜁) in Frobenius norm. Manipulating, ˆ 𝐷ˆ − 𝐼) 𝑆ˆ⊤ ∥ 𝐹 ≤ ∥ 𝑆∥ ˆ 2op ∥ 𝐷ˆ − 𝐼 ∥ 𝐹 = ∥ 𝑆ˆ 𝑆ˆ⊤ ∥ op ∥ 𝐷ˆ − 𝐼 ∥ 𝐹 ∥𝑉ˆraw − 𝑉ˆ ∥ 𝐹 = ∥ 𝑆(

(89)

We do this by taking inspiration from [CMF+ 26] and introducing the auxiliary matrix 𝐴(𝑊) ≡ −Ω𝑊Ω𝑊, where 𝑊 is any real positive-definite matrix. Since 𝑆ˆ and 𝑆 are symplectic, Ω𝑆ˆ = 𝑆ˆ −⊤ Ω and Ω𝑆 = 𝑆 −⊤ Ω. Then, ˆ 𝐷ˆ 𝑆ˆ⊤ = 𝑆ˆ −⊤ 𝐷ˆ 2 𝑆ˆ⊤ 𝐴(𝑉ˆraw ) = −Ω𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ Ω𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ = −𝑆ˆ −⊤ Ω 𝐷Ω ˆ Ω share the same block structure and commute. A similar where we use the fact that Ω2 = −𝐼 and 𝐷, −⊤ 2 ⊤ calculation gives that 𝐴(𝑉) = 𝑆 𝐼 𝑆 = 𝐼, meaning 𝐴(𝑉ˆraw ) − 𝐴(𝑉) = 𝑆ˆ −⊤ ( 𝐷ˆ 2 − 𝐼) 𝑆ˆ⊤ . Then, writing the symplectic eigenvalues of 𝑉ˆraw , 𝐷ˆ = diag(𝜆ˆ 1 , . . . , 𝜆ˆ 𝑛 , 𝜆ˆ 1 , . . . , 𝜆ˆ 𝑛 ) and both 𝐴(𝑉ˆraw ) − 𝐴(𝑉) and 𝐷ˆ 2 − 𝐼 have

68

the eigenvalues {𝜆ˆ 21 − 1, . . . , 𝜆ˆ 2𝑛 − 1, 𝜆ˆ 21 − 1, . . . , 𝜆ˆ 2𝑛 − 1}. Then, since 𝑉ˆraw ≻ 0 implies 𝜆ˆ 𝑖 > 0, ∥ 𝐷ˆ − 𝐼 ∥ 2𝐹 = 2

𝑛 ∑︁

(𝜆ˆ 𝑖 − 1) 2 ≤ 2

𝑖=1

𝑛 ∑︁

(𝜆ˆ 2𝑖 − 1) 2 =

𝑖=1

2𝑛 ∑︁

|𝜆 𝑗 ( 𝐴(𝑉ˆ0 ) − 𝐴(𝑉))| 2

𝑗=1

≤ ∥ 𝐴(𝑉ˆraw ) − 𝐴(𝑉) ∥ 2𝐹 ≤ ∥ − Ω(𝑉ˆraw − 𝑉)Ω𝑉 − Ω𝑉Ω(𝑉ˆraw − 𝑉) − Ω(𝑉ˆraw − 𝑉)Ω(𝑉ˆraw − 𝑉) ∥ 2𝐹 ≤ (2∥𝑉 ∥ op ∥𝑉ˆraw − 𝑉 ∥ 𝐹 + ∥𝑉ˆraw − 𝑉 ∥ 2𝐹 ) 2 ≤ 92 𝜁 2

(90)

Í Here, the first line uses the elementary facts that |𝑥 − 1| ≤ |𝑥 2 − 1| for any 𝑥 > 0 and 𝑗 |𝜆 𝑗 (𝑀)| 2 ≤ ∥ 𝑀 ∥ 2𝐹 for any square matrix 𝑀. The remainder of the manipulation proceeds from the prior absolute bound ∥𝑉ˆraw − 𝑉 ∥ 𝐹 ≤ 2𝜁 and 𝜁 < 1/4. We now switch our focus to the first term of (89). Taking 𝜁 < 1/18, from (90), 𝐷ˆ ⪰ (1 − 9𝜁)𝐼 ⪰ 𝐼/2 and 𝑉ˆraw ⪯ 𝑉 + 2𝜁 𝐼 ≤ 3𝐼, from ∥𝑉ˆraw − 𝑉 ∥ 𝐹 ≤ 2𝜁 and 𝑉 ⪯ 2𝐼. Then, 𝑆ˆ 𝑆ˆ⊤ ≤ 2𝑆ˆ 𝐷ˆ 𝑆ˆ⊤ = 2𝑉ˆraw ⪯ 6𝐼

(91)

Substituting (91) and (90) into (89), we have that ∥𝑉ˆraw − 𝑉ˆ ∥ 𝐹 ≲ 𝜁. Then, by triangle inequality, ∥𝑉ˆ − 𝑉 ∥ 𝐹 ≤ ∥𝑉ˆ − 𝑉ˆraw ∥ 𝐹 + ∥𝑉ˆraw − 𝑉 ∥ 𝐹 ≲ 𝜁 + 2𝜁 ≲ 𝜁 meaning ∥𝑉 −1/2 (𝑉ˆ − 𝑉)𝑉 −1/2 ∥ 𝐹 ≲ 𝜁 since 𝐼/2 ⪯ 𝑉. Then, this relative bound on the covariance combined ˆ with the relative bound on the mean in (88) gives that 𝑑tv (N (m, 𝑉/2), N ( m̂, 𝑉/2)) ≲ 𝜁 as well. Applying + Lemma B.3 of [CMF 26] to translate this closeness in phase space to trace distance closeness between the two states, ˆ ˆ 𝜓|) ˆ ≤ 150 · 𝑑tv (N (m, 𝑉/2), N ( m̂, 𝑉/2)) 𝑑tr (|𝜓⟩⟨𝜓| , | 𝜓⟩⟨ ≲𝜁 provided 𝜁 is sufficiently small such that this total variation distance is less than 1/600. A.2.4

□

Proof of Lemma 3.11

Proof. WLOG, by multilinearity, ℎ1 , . . . , ℎ 𝑘 each have unit norm ∥ · ∥ P = 1. We work with the unnormalized vector |Φ(𝜃)⟩ = 𝑒 𝐾 𝜃 |0⟩ for now. Since 𝐾 𝜃 is linear in 𝜃 and all creation operators commute, 𝑒 𝐾 𝜃+𝑡 ℎ = 𝑒 𝐾 𝜃 +𝑡 𝐾ℎ = 𝑒 𝐾 𝜃 𝑒 𝑡 𝐾ℎ 𝑑 𝑒 𝐾 𝜃+𝑡 ℎ1 |0⟩ = 𝐾 ℎ1 𝑒 𝐾 𝜃 |0⟩ 𝑑𝑡 𝑡=0 𝑗 ∞ ∑︁ 𝐾 ℎ1 · · · 𝐾 ℎ𝑘 𝐾 𝜃 |0⟩ 𝑘 𝐾𝜃 𝐷 Φ(𝜃) [ℎ1 , . . . , ℎ 𝑘 ] = 𝐾 ℎ1 · · · 𝐾 ℎ𝑘 𝑒 |0⟩ = 𝑗! 𝑗=0 |𝐷Φ(𝜃) [ℎ1 ]⟩ =

69

(92)

Focusing on bounding this numerator, we sequentially apply Lemma 3.10 to the 𝑗 + 𝑘 operators in the product, recalling that each term increases the total photon count by at most two: 𝑗 ∥𝐾 ℎ1 · · · 𝐾 ℎ𝑘 𝐾 𝜃 |0⟩ ∥ 2 ≤

𝑗+𝑘−1 Ö

=4

𝑠=0 𝑗+𝑘

! 2(2𝑠 + 2)

𝑗 ∥𝜃 ∥ P

𝑘 Ö

 𝑖 ∥ P ∥ℎ 

𝑖=1  𝑗 ( 𝑗 + 𝑘)!∥𝜃 ∥ P

(93)

Applying triangle inequality to (92) and then using (93) to bound each term, since ∥𝜃 ∥ P ≤ 1/8, ∥ 𝐷 𝑘 Φ(𝜃) [ℎ1 , . . . , ℎ 𝑘 ] ∥ 2 ≤ 4 𝑘 𝑘!

 ∞  ∑︁ 𝑗+𝑘 𝑘

(4∥𝜃 ∥ P ) 𝑗 =

𝑗=0

4 𝑘 𝑘! ≤ 2 · 8𝑘 𝑘 (1 − 4𝑟) 𝑘+1

(94)

Í 𝑗+𝑘
𝑗 −𝑘−1 for |𝑥| < 1. where the equality uses the generalized binomial theorem ∞ 𝑗=0 𝑘 𝑥 = (1 − 𝑥) We now revive the normalization 𝑍 (𝜃) = ⟨Φ(𝜃)|Φ(𝜃)⟩ and use the multivariable product rule for the Frèchet derivative: |𝐷 𝑘 𝑍 (𝜃) [ℎ1 , . . . , ℎ 𝑘 ] | =

∑︁

𝐷 |𝑆 | Φ(𝜃) [ℎ𝑖 : 𝑖 ∈ 𝑆] 𝐷 𝑘− |𝑆 | Φ(𝜃) [ℎ𝑖 : 𝑖 ∉ 𝑆]

𝑆⊆[𝑘]

≤

𝑘   ∑︁ 𝑘

𝐷 𝑗 Φ(𝜃)

𝑗

op

𝐷 𝑘− 𝑗 Φ(𝜃)

op

𝑗=0

≤

𝑘   ∑︁ 𝑘

(2 · 8 𝑗 𝑗!) (2 · 8 𝑘− 𝑗 (𝑘 − 𝑗)!)

𝑗 𝑗=0

= 4 · 8 𝑘 (𝑘 + 1)!

(95)

where we group by |𝑆| = 𝑗 in the second line after triangle inequality and also apply (94) in the third line, with ∥ · ∥ op denoting the operator norm with respect to ∥ · ∥ P . We now analyze 𝑁 (𝜃) = 𝑔 ◦ 𝑍 where 𝑔(𝑥) = 𝑥 −1/2 . Since 𝑍 ≥ 1 from the vacuum term, we restrict to Î 𝑥 ≥ 1 and can bound |𝑔 (𝑚) (𝑥)| = 𝑥 −𝑚−1/2 𝑚−1 𝑗=0 ( 𝑗 + 1/2) ≤ 𝑚!. Then, applying the higher-order chain rule

70

for the Frèchet derivative [CS96] along with (95) and |𝑔 (𝑚) (𝑥)| ≤ 𝑚!, |𝐷 𝑘 𝑁 (𝜃) [ℎ1 , . . . , ℎ 𝑘 ] | =

∑︁ S ∈𝔓𝑘

≤

|𝑔 | S ∈𝔓 𝑘 ∑︁

Ö ∥𝐷 |𝑆 | 𝑍 (𝜃) ∥ op (𝑍 (𝜃))| {z } 𝑆∈S | {z }

(|S|)

≤ 16 𝑘

≤ | S |!

∑︁

4𝑗 𝑘 ∑︁ 𝑗=1

4·8 |𝑆| ( |𝑆 |+1)!

𝑗!

S ∈𝔓𝑘 :| S |= 𝑗

𝑗=1

𝐷 |𝑆 | 𝑍 (𝜃) [ℎ𝑖 : 𝑖 ∈ 𝑆]

𝑆∈S

∑︁

𝑘

≤

Ö

𝑔 ( | S | ) (𝑍 (𝜃))

Ö 𝑆∈S

∑︁

4𝑗

𝑗!

S ∈𝔓𝑘 :| S |= 𝑗

|

16 |𝑆 | |𝑆|! Ö

|𝑆|!

𝑆∈S

{z 𝑘! ( 𝑘−1 𝑗 −1 )

}

≤ 16 𝑘 𝑘! · 4 · 5 𝑘−1 ∥𝐷 𝑘 𝑁 (𝜃) ∥ op ≤ 80 𝑘 𝑘!

(96)

where 𝔓𝑘 is the set of partitions of [𝑘]. The bracketed term in the fourth line simplifies by noticing that both quantities count the number of ways to group 𝑘 distinguishable elements into 𝑗 groups, either by enumerating over all partitions of size 𝑗 and permuting within each group and then permuting the groups themselves, or alternatively permuting all the elements and then choosing the 𝑗 − 1 cuts that define the groups. The fifth Í 𝑘 𝑘−1
𝑟 𝑘−1 with 𝑥 = 4. line follows from the binomial theorem 𝑟=1 𝑟 −1 𝑥 = 𝑥(1 + 𝑥) Then, since |𝜓(𝜃)⟩ = 𝑁 |Φ(𝜃)⟩, we apply the multivariable product rule one last time: ∥ 𝐷 𝑘 𝜓(𝜃) [ℎ1 , . . . , ℎ 𝑘 ] ∥ 2 =

∑︁

𝐷 |𝑆 | 𝑁 (𝜃) [ℎ𝑖 : 𝑖 ∈ 𝑆] · 𝐷 𝑘− |𝑆 | Φ(𝜃) [ℎ𝑖 : 𝑖 ∉ 𝑆]

𝑆⊆[𝑘]

≤

2

𝑘   ∑︁ 𝑘

∥𝐷 𝑗 𝑁 (𝜃) ∥ op ∥𝐷 𝑘− 𝑗 Φ(𝜃) ∥ op 𝑗 | {z } | {z } 𝑗=0 ≤80 𝑗 𝑗!

≤2·8 𝑘− 𝑗 (𝑘− 𝑗 )!

𝑘

≤ 2 · 88 𝑘! where we use (94), (96), and binomial theorem for (80 + 8) 𝑘 . Thus, the lemma holds for 𝐶𝐷 = 176 or better. □ A.2.5

Proof of Lemma 3.13

√ Proof. Let 𝑒 1 , . . . , 𝑒 𝑑P be a real orthonormal basis for P, where 𝑑 P = 𝑛2 + 3𝑛. Taking ℎ = 𝜅/(3 𝑑), estimate 𝑓𝜌 (·) at each of 𝜃, 𝜃 + ℎ𝑒 1 , . . . , 𝜃 + ℎ𝑒 𝑑 in 𝐵 𝑅 to accuracy 𝜅0 ≡ 𝜅 2 /(12𝑑) and failure probability

71

𝛿/(𝑑 + 1). Then, construct the zeroth-order estimator: d ∇ 𝑓𝜌 (𝜃) =

𝑑 ˆ ∑︁ 𝑓𝜌 (𝜃 + ℎ𝑒 𝑖 ) − 𝑓ˆ𝜌 (𝜃)

𝑒𝑖 .

ℎ 𝑖=1

Then, by the fundamental theorem of calculus and (23), ∫ ℎ 𝑓𝜌 (𝜃 + ℎ𝑒 𝑖 ) = 𝑓𝜌 (𝜃) + ℎ𝐷 𝑓𝜌 (𝜃) [𝑒 𝑖 ] + (ℎ − 𝑡)𝑔 ′′ (𝑡)𝑑𝑡 0 ∫ ℎ ∫ ℎ 𝑓𝜌 (𝜃 + ℎ𝑒 𝑖 ) − 𝑓𝜌 (𝜃) 1 3 3ℎ 2 − 𝐷 𝑓𝜌 (𝜃) [𝑒 𝑖 ] = (ℎ − 𝑡)𝐷 𝑓𝜌 (𝜃 + 𝑡𝑒 𝑖 ) [𝑒 𝑖 , 𝑒 𝑖 ]𝑑𝑡 ≤ (ℎ − 𝑡)𝑑𝑡 = ℎ ℎ 0 ℎ 0 2 Since estimating 𝑓𝜌 (·) contributes at most 2𝜅0 /ℎ error to the zeroth-order estimate in each coordinate, by triangle inequality,   √ 3ℎ 2𝜅 0 d ∥ ∇ 𝑓𝜌 (𝜃) − ∇ 𝑓𝜌 (𝜃) ∥ P ≤ 𝑑 + = 𝜅. 2 ℎ The copy complexity and failure probability are immediate.

A.3

Deferred proofs from Section 4

A.3.1

Proof of Lemma 4.3

□

Proof. The true precision 𝐾 = Σ −1 belongs to K 𝐴. Moreover, Σ ⪰ 𝐼/2, Σ̂ ⪰ Σ/𝑡 𝛼 , and ∥ Σ̂ − Σ∥ 𝐹 ≤ 𝐴𝑡 2𝛼 . The resolvent identity therefore gives ∥ 𝐾ˆ − 𝐾 ∥ 𝐹 ≤ ∥ Σ̂ −1 ∥ op ∥ Σ̂ − Σ∥ 𝐹 ∥Σ −1 ∥ op ≲ 𝐴𝑡 3𝛼 . Projection optimality and the triangle inequality imply ∥𝐾0 − 𝐾 ∥ 𝐹 ≲ 𝐴𝑡 3𝛼 . The spectral constraints give ∥𝐾0−1 ∥ op ≲ 𝐴 and ∥𝑉0±1 ∥ op ≲ 𝐴. Since ∥𝐾 −1 ∥ op = ∥Σ∥ op ≤ 𝐴, we obtain ∥𝑉0 − 𝑉 ∥ 𝐹 = 2∥𝐾0−1 (𝐾 − 𝐾0 )𝐾 −1 ∥ 𝐹 ≲ 𝐴3 𝑡 3𝛼 , ∥𝑍 − 𝐼 ∥ 𝐹 = ∥𝑆0−1 (𝑉 − 𝑉0 )𝑆0−⊤ ∥ 𝐹 ≲ 𝐴4 𝑡 3𝛼 ,

𝑍 = 𝑆0−1𝑉 𝑆0−⊤ .

For the transformed mean 𝑒 = 𝑆0−1 (m − 𝜇), ˆ the mean guarantee similarly gives ∥𝑒∥ 22 ≤ ∥𝑉0−1 ∥ op ∥m − 𝜇∥ ˆ 22 ≲ 𝐴2 𝑡 𝛼 (1 + 𝑡 𝛼 ). The eigenvalues of the pure covariance 𝑍 occur in reciprocal pairs. For each pair, writing 𝑥 = 𝜆 + 𝜆 −1 − 2 ≥ 0 gives (𝜆 − 1) 2 + (𝜆 −1 − 1) 2 = 𝑥 2 + 2𝑥 ≥ 2𝑥. Thus tr 𝑍 − 2𝑛 ≤ ∥𝑍 − 𝐼 ∥ 2𝐹 /2. The transformed photon number is therefore bounded by 𝜈=

tr 𝑍 − 2𝑛 ∥𝑒∥ 22 + ≲ 𝑡 6𝛼 𝐴8 + 𝑡 𝛼 (1 + 𝑡 𝛼 ) 𝐴2 ≤ 𝑢 𝛼 𝐴8 . 4 2 72

For example, the preceding estimates give ∥𝑍 − 𝐼 ∥ 𝐹 ≤ 96𝐴4 𝑡 3𝛼 and ∥𝑒∥ 22 ≤ 3𝐴2 𝑡 𝛼 (1 + 𝑡 𝛼 ), so the stated value of 𝑢 𝛼 suffices. □ A.3.2

Proof of Lemma 4.4

Proof. Conjugating 𝑁𝐺 + 𝑁 𝐻 by 𝐺 gives 𝑁 + 𝑈𝑁𝑈 † , where 𝑈𝑁𝑈 † =

1 𝑛 ( R̂ − d) ⊤𝐶 −1 ( R̂ − d) − 𝐼. 2 2

Set 𝑀 = 𝐼 + 𝐶 −1 ,

𝑢 = 𝑀 −1𝐶 −1 d = (𝐼 + 𝐶) −1 d.

Expanding the quadratic terms and using 𝑀𝑢 = 𝐶 −1 d gives 1 1 𝑁 + 𝑈𝑁𝑈 † = R̂⊤ 𝑀 R̂ − d⊤𝐶 −1 R̂ + d⊤𝐶 −1 d − 𝑛𝐼 2 2 1 1 = ( R̂ − 𝑢) ⊤ 𝑀 ( R̂ − 𝑢) + d⊤ [𝐶 −1 − 𝐶 −1 𝑀 −1𝐶 −1 ]d − 𝑛𝐼 2 2 1 1 ⊤ = ( R̂ − 𝑢) 𝑀 ( R̂ − 𝑢) + d⊤ (𝐼 + 𝐶) −1 d − 𝑛𝐼. 2 2 Here scalar terms multiply the identity. For the last equality, use 𝑀 −1 = (𝐼 + 𝐶) −1𝐶 and 𝐼 − (𝐼 + 𝐶) −1 = 𝐶 (𝐼 + 𝐶) −1 to obtain 𝐶 −1 − 𝐶 −1 𝑀 −1𝐶 −1 = 𝐶 −1 [𝐼 − (𝐼 + 𝐶) −1 ] = (𝐼 + 𝐶) −1 . A displacement removes 𝑢 without changing the spectrum. A passive change of modes diagonalizes the pure covariance 𝐶 into reciprocal pairs (𝑒 2𝑟 𝑗 , 𝑒 −2𝑟 𝑗 ). The corresponding symplectic eigenvalues of 𝑀 = 𝐼 + 𝐶 −1 are √︃ 𝜔 𝑗 = (1 + 𝑒 2𝑟 𝑗 ) (1 + 𝑒 −2𝑟 𝑗 ) = 2 cosh 𝑟 𝑗 . The Gaussian transformation obtained from the Williamson decomposition of 𝑀 turns the centered quadratic Í Í Í Hamiltonian into 𝑗 𝜔 𝑗 ( 𝑎ˆ †𝑗 𝑎ˆ 𝑗 + 1/2), whose ground energy is 12 𝑗 𝜔 𝑗 = 𝑗 cosh 𝑟 𝑗 . Adding the scalar terms above proves (43). □ A.3.3

Proof of Lemma 4.10

Proof. The domain is the intersection of three norm balls in the real vector space P𝑛 , so it is convex. Its operator-norm constraint ensures ∥ 𝐴∥ op < 1, as required for the Bargmann state. Suppose first that 𝜈(𝜋 𝑏, 𝐴) ≤ ℎ. Each term in (9) is nonnegative, so ∥ 𝛽∥ 22 ≤ ℎ and 𝑠2𝑗 /(1 − 𝑠2𝑗 ) ≤ ℎ for every 𝑗. Rearranging the latter inequality gives 𝑠2𝑗 ≤ ℎ/(ℎ + 1), and therefore ∥ 𝐴∥ op ≤ 𝑟 ℎ . Moreover, ∥ 𝐴∥ 2𝐹 =

∑︁ 𝑗

𝑠2𝑗 ≤

∑︁

𝑠2𝑗

𝑗

1 − 𝑠𝑗

√ ∥𝑏∥ 2 ≤ (1 + ∥ 𝐴∥ op ) ∥ 𝛽∥ 2 ≤ 2 ℎ.

≤ ℎ, 2

These are precisely the three constraints defining Dℎ .

73

Conversely, suppose (𝑏, 𝐴) ∈ Dℎ . Since 𝑠 𝑗 ≤ 𝑟 ℎ and 1 − 𝑟 ℎ2 = 1/(ℎ + 1), ∑︁

𝑠2𝑗

𝑗

1 − 𝑠𝑗

2 𝑗 𝑠𝑗

∥ 𝐴∥ 2𝐹

1 − 𝑟ℎ

1 − 𝑟 ℎ2

Í ≤ 2

= 2

≤ ℎ(ℎ + 1).

Moreover, (8) and the identity 1 + 𝑟ℎ 1 = (1 + 𝑟 ℎ ) (ℎ + 1) ≤ 2(ℎ + 1) = 1 − 𝑟 ℎ 1 − 𝑟 ℎ2 give ∥ 𝛽∥ 22 ≤

∥𝑏∥ 22 (1 − 𝑟 ℎ ) 2

≤ 16ℎ(ℎ + 1) 2 .

The conclusion immediately follows. A.3.4

□

Proof of Lemma 4.13

Proof. Estimate matrix entries Let 𝐷 = 𝐷 𝐾 and let |𝑒 1 ⟩ , . . . , |𝑒 𝐷 ⟩ be the canonical orthonormal basis for the cutoff space. For each 𝑖 measure 𝐸 𝑖 = |𝑒 𝑖 ⟩ ⟨𝑒 𝑖 |, and for each 𝑖 < 𝑗 measure 𝑅𝑖 𝑗 =

|𝑒 𝑖 ⟩ 𝑒 𝑗 + 𝑒 𝑗 ⟨𝑒 𝑖 | , 2

𝐽𝑖 𝑗 =

i |𝑒 𝑖 ⟩ 𝑒 𝑗 − i 𝑒 𝑗 ⟨𝑒 𝑖 | . 2

The expectations of these observables on 𝜎 are, respectively, (𝜎𝐾 )𝑖𝑖 , Re(𝜎𝐾 )𝑖 𝑗 , and Im(𝜎𝐾 )𝑖 𝑗 . Their measurement outcomes lie in [−1, 1].
There are 𝐷 + 2 𝐷2 = 𝐷 2 quantities to estimate. Let 𝛿 = 𝜖/(64𝐷) and measure each observable on l  m 𝑚 = 2𝛿 −2 log 2𝐷 2 /𝜂 2

fresh copies. Hoeffding’s inequality bounds the error probability for each empirical mean by 2𝑒 −𝑚 𝛿 /2 ≤ 𝜂/𝐷 2 , so by the union bound all errors are at most 𝛿 with probability at least 1 − 𝜂. Let 𝜎 ˜ 𝐾 be defined using these empirical estimates, and make it Hermitian by taking conjugates. On this event the diagonal errors have magnitude at most 𝛿, and the squared magnitude of each off-diagonal error is at most 2𝛿2 . Hence ∥𝜎 ˜ 𝐾 − 𝜎𝐾 ∥ 2𝐹 ≤ 𝐷𝛿2 + 2𝐷 (𝐷 − 1)𝛿2 ≤ 2𝐷 2 𝛿2 ,

∥𝜎 ˜ 𝐾 − 𝜎𝐾 ∥ 𝐹 ≤ 𝜖/32.

The copy count is 𝐷 2 𝑚 = 𝑂 (𝐷 4 𝜖 −2 log(2𝐷/𝜂)). Project onto set of subnormalized states Let 𝜎 ˆ 𝐾 be a nearest matrix to 𝜎 ˜ 𝐾 in Frobenius norm in the closed convex set S𝐷 = {𝑋 = 𝑋 † : 𝑋 ⪰ 0, tr 𝑋 ≤ 1}. It is not hard to see that this projection can be computed by diagonalizing 𝜎 ˜ 𝐾 and projecting its eigenvalues Í 𝐷 ˜ 𝐾 is at most 𝜖/32 from S𝐷 , so 𝜎 ˆ 𝐾 is at most 𝜖/16 onto {𝑥 ∈ R : 𝑥 𝑖 ≥ 0, 𝑖 𝑥 𝑖 ≤ 1}. Since 𝜎𝐾 ∈ S𝐷 , 𝜎 from 𝜎𝐾 . □

74

A.4

Deferred proofs from Section 5

A.4.1

Proof of Lemma 5.7

Proof. From Lemma 7 of [WY19] and 2.11.1 of [Tim14], there exist two unit mean random variables on {0} ∪ [1 + 𝜈, 𝜆]who match the first 𝑘 moments and have: 

√︃

2

√︃ 𝑘 2 1+𝜈 © ª 𝜆 ­1 − ® Pr[𝑈+ > 0] − Pr[𝑈− > 0] = inf sup |1/𝑥 − 𝑝(𝑥)| = √︃ ­ ® 𝑝∈ P𝐿 𝑥 ∈ [1,𝐶 𝑘 2 ] 1+𝜈 1+𝜈 1+ 𝜆 « ¬ 𝑘  2 = (1 + 𝜆 −1/2 ) 2 1 − √ 1+ 𝜆 1+

1+𝜈 𝜆

where P𝐿 is the set of real polynomials of degree at most 𝐿 and where we take 𝜈 = 0 and 𝜆 = 𝐶 𝑘 2 . Bounding, Pr[𝑈+ > 0] − Pr[𝑈− > 0] = (1 + 𝜆

−1/2 2

)



2 1− √ 1+ 𝜆

𝑘 ≥1−

2 2𝑘 √ ≥1− √ ≡ 𝑐 1+ 𝜆 𝐶

for sufficiently large 𝐶.

A.5

Deferred proofs from Section 6

A.5.1

Proof of Lemma 6.3

□

Proof. We begin with the one-mode setting where 𝑏, 𝐴 ∈ C. Since 𝑝 1 (|𝜓⟩) does not change if we rotate † |𝜓⟩ ↦→ 𝑒 −𝑖 𝜃 𝑎ˆ 𝑎ˆ |𝜓⟩ = |𝜓(𝑒 −𝑖 𝜃 𝑏, 𝑒 −2𝑖 𝜃 𝐴)⟩, WLOG, we can take 𝑏 > 0 and 𝐴 ∈ C. Identifying 𝑏 = 𝛽 and 𝐴 = 𝛼, we see then that all pure Gaussian states are in the family, and using (6) and (80),  2      √︁ √︁ 𝛽2 𝛽 (1 + Re 𝛼) 𝛽2 (1 − |𝛼|) 2 2 2 2 = 𝛽 1 − |𝛼| exp − 𝑝 1 (|𝜓⟩) = ≤ 𝛽 1 − |𝛼| exp − (97)   𝑍 (𝛽, 𝛼) (1 − |𝛼|) (1 + |𝛼|) 1 − |𝛼| 2  √︁ ≤ 𝑒 −1 (1 + |𝛼|) 1 − |𝛼| 2 ≤ 𝑝 ∗1 where the first inequality uses Re 𝛼 ≥ −|𝛼|, the second√inequality uses the fact that 𝑡𝑒 −𝑡/𝑎 is maximized at 𝑡 = 𝑎, and third inequality uses the fact that (1 + 𝑡) 1 − 𝑡 2 is maximized at 𝑡 = 1/2. Then, equality throughout gives that Re 𝛼 = −|𝛼| = −1/2 and 𝛽2 = 1 + |𝛼| = 3/2, meaning (𝛽, 𝛼) = (𝛽∗ , 𝛼∗ ). We now show the desired stability bound near the maximizer, beginning with stability in parameter distance. Packaging 𝜃 = (𝛽, Re 𝛼, Im 𝛼) ∈ R3 and 𝜃 ∗ = (𝛽∗ , 𝛼∗ , 0), define the objective: − log 𝑝 1 (|𝜓⟩) ≡ 𝑓 (𝜃) = −2 log 𝜃 1 −

  𝛽2 (1 + 𝜃 ) 1 2 log 1 − 𝜃 22 − 𝜃 32 + 2 1 − 𝜃 22 − 𝜃 32

√  8/3 4 6/9 0  √   Differentiating, one can verify that ∇ 𝑓 (𝜃 ∗ ) = 0, ∇2 𝑓 (𝜃 ∗ ) = 4 6/9 28/9 0 ⪰ 16 9 𝐼, and every entry     0 0 4  75

of ∇2 𝑓 (𝜃) is continuous when 𝜃 1 > 0 and 𝜃 22 + 𝜃 32 < 1. Then, by continuity of the Hessian and 𝑝(𝜃), there exists a universal constant 𝜉 for which any 𝜃 with ∥𝜃 − 𝜃 ∗ ∥ 2 < 𝜉 has ∇2 𝑓 (𝜃) ⪰ 𝐼 and 𝑝 1 (𝜃) ≥ 𝑝 ∗1 /2. For now, restrict 𝜃 to this ball. Using this and the fundamental theorem of calculus applied twice, 𝑓 (𝜃) − 𝑓 (𝜃 ∗ ) =

∫ 1

(1 − 𝑡) (𝜃 − 𝜃 ∗ ) ⊤ ∇2 𝑓 (𝜃 ∗ + 𝑡 (𝜃 − 𝜃 ∗ )) (𝜃 − 𝜃 ∗ )𝑑𝑡 ≥

0

∫ 1 0

(1 − 𝑡) ∥𝜃 − 𝜃 ∗ ∥ 22 𝑑𝑡

1 ≥ ∥𝜃 − 𝜃 ∗ ∥ 22 2 This then gives stability in parameter distance: |𝛽 − 𝛽∗ | 2 + |𝛼 − 𝛼∗ | 2 ≤ 𝑓 (𝜃) − 𝑓 (𝜃 ∗ ) = log

𝑝 ∗ − 𝑝 1 (|𝜓⟩) 𝑝 ∗1 ≤ 1 ≲ 𝑝 ∗1 − 𝑝 1 (|𝜓⟩) 𝑝 1 (|𝜓⟩) 𝑝 1 (|𝜓⟩)

(98)

where use the fact that log 𝑡 ≤ 𝑡 − 1 and 𝑝 1 (𝜃) ≥ 𝑝 ∗1 /2. We now convert this parameter distance control into stability in infidelity. From (84), we have that ( 𝑎ˆ − 𝛼∗ 𝑎ˆ † − 𝛽∗ ) |𝜓 ∗ ⟩ = 0 where [ 𝑎ˆ − 𝛼∗ 𝑎ˆ † − 𝛽∗ , 𝑎ˆ † − 𝛼∗ 𝑎ˆ − 𝛽∗ ] = (1 − (𝛼∗ ) 2 )𝐼. Then, define the operator: 𝑎ˆ − 𝛼∗ 𝑎ˆ † − 𝛽∗ 𝑎ˆ 𝛼∗ ,𝛽∗ ≡ √︁ 1 − (𝛼∗ ) 2 such that 𝑎ˆ 𝛼∗ ,𝛽∗ |𝜓 ∗ ⟩ = 0 and [ 𝑎ˆ 𝛼∗ ,𝛽∗ , 𝑎ˆ †𝛼∗ ,𝛽∗ ] = 1, meaning. Since this operator is just formed by squeezing and displacement, 𝑎ˆ 𝛼∗ ,𝛽∗ = 𝑈 𝑎𝑈 ˆ † for some Gaussian unitary 𝑈 where 𝑈 |0⟩ = |𝜓 ∗ ⟩. Then, since 𝐼 − |0⟩⟨0| ⪯ † 𝑎ˆ 𝑎, ˆ conjugating by 𝑈 gives 𝐼 − |𝜓 ∗ ⟩⟨𝜓 ∗ | ⪯ 𝑎ˆ †𝛼∗ ,𝛽∗ 𝑎ˆ 𝛼∗ ,𝛽∗ . Considering the infidelity, 𝑑IF (|𝜓⟩ , |𝜓 ∗ ⟩) = ⟨𝜓| (𝐼 − |𝜓 ∗ ⟩⟨𝜓 ∗ |) |𝜓⟩ ≤ ∥ 𝑎ˆ 𝛼∗ ,𝛽∗ |𝜓⟩ ∥ 22 =

4 ∥ (𝛽 − 𝛽∗ ) |𝜓⟩ + (𝛼 − 𝛼∗ ) 𝑎ˆ † |𝜓⟩ ∥ 22 3

8 |𝛽 − 𝛽∗ | 2 + |𝛼 − 𝛼∗ | 2 ∥ 𝑎ˆ † |𝜓⟩ ∥ 22 3 ≲ ( 𝑝 ∗1 − 𝑝 1 (|𝜓⟩)) (1 + ∥ 𝑎ˆ † |𝜓⟩ ∥ 22 ) ≤

where the second inequality uses the fact ∥𝑢 + 𝑣∥ 22 ≤ 2∥𝑢∥ 22 + 2∥𝑣∥ 22 and the final bound applies (98). Then, since 𝑎ˆ 𝑎ˆ † = 𝑎ˆ † 𝑎ˆ + 𝐼, √︃ † ∥ 𝑎ˆ |𝜓⟩ ∥ 2 = ∥ 𝑎ˆ |𝜓⟩ ∥ 22 + 1 ≤ ∥ 𝑎ˆ |𝜓⟩ ∥ 2 + 1 = ∥ (𝛼 𝑎ˆ † + 𝛽) |𝜓⟩ ∥ 2 + 1 ≤ 𝛽 + |𝛼|∥ 𝑎ˆ † |𝜓⟩ ∥ 2 + 1 1+𝛽 ∗ meaning ∥ 𝑎ˆ † |𝜓⟩ ∥ 2 ≤ 1− | 𝛼| ≲ 1 since 𝛼 is bounded away from 1 by ∥𝜃 − 𝜃 ∥ < 𝜉. Thus, there exists a ∗ ∗ ∗ constant 𝐶 for which 𝑑IF (|𝜓⟩ , |𝜓 ⟩) ≤ 𝐶 ( 𝑝 1 − 𝑝 1 (|𝜓⟩)) when ∥𝜃 − 𝜃 ∥ < 𝜉. Then, to finish the claim, it suffices to show that for ∥𝜃 − 𝜃 ∗ ∥ 2 ≥ 𝜉, 𝑝 ∗1 − 𝑝 1 (|𝜓⟩) ≥ 𝛾 for some 𝛾 > 0 so that 𝑑IF (|𝜓⟩ , |𝜓 ∗ ⟩) ≤ 𝐶 ′ ( 𝑝 ∗1 − 𝑝 1 (|𝜓⟩)) for 𝐶 ′ = max{𝐶, 1/𝛾}. Outside the ball, if 𝑝 1 < 𝑝 ∗1 /2, we are already bounded away. Further, from the bounding (97), it is clear that {𝜃 : 𝑝 1 (|𝜓(𝜃)⟩) ≥ 𝑝 ∗1 /2} is compact. Then, the intersection of this set with the complement of the ball is also compact, making 𝜃 ∗ the unique minimizer on this region, which finishes the one-mode case. We now generalize to arbitrary modes. Consider 𝑑IF (|𝜓⟩ , |𝜓 ∗ (𝑥)⟩) for 𝑥 = 𝑏/∥𝑏∥. Then, applying a

76

passive unitary, by the unitary invariance of fidelity, 𝑑IF (|𝜓(𝑏, 𝐴)⟩ , |𝜓 ∗ (𝑥)⟩) = 𝑑IF (|𝜓 ′ (𝑏 ′ , 𝐴′ )⟩ , |𝜓 ∗ (𝑒 1 )⟩)

(99)

where 𝑏 ′ = ∥𝑏∥ 2 𝑒 1 = 𝑊 𝑏 for unitary matrix 𝑊 and 𝐴′ = 𝑊 𝐴𝑊 ⊤ . Looking at (5),   𝛼∗ † 2 ∗ ∗ † |𝜓 (𝑒 1 )⟩ = exp 𝛽 𝑎ˆ 1 + ( 𝑎ˆ ) |0⟩ ⊗𝑛 = |𝜓 ∗ ⟩ ⊗ |0⟩ ⊗ (𝑛−1) 2 1 where |𝜓 ∗ ⟩ is the single-mode state in the family that maximizes 𝑝 1 (·). Then, considering the projection of |𝜓 ′ ⟩ onto the first mode given by projector 𝑃𝑒1 = 𝐼 ⊗ |0⟩⟨0| ⊗ (𝑛−1) which projects onto first-mode states, ′

𝑃𝑒1 |𝜓 ⟩ = √︁



1 𝑍 (𝑏 ′ , 𝐴′ )

exp

 ′ 𝐴11 † † 2 ∥𝑏∥ 2 𝑎ˆ 1 + ( 𝑎ˆ 1 ) |0⟩ ⊗𝑛 = 𝜅 |𝜓(𝛽, 𝛼)⟩ ⊗ |0⟩ ⊗ (𝑛−1) 2

2 ′ for 𝜅 = 𝑍𝑍(𝑏(𝛽,′ , 𝛼) 𝐴′ ) = ∥𝑃𝑒1 |𝜓 ⟩ ∥ 2 ≤ 1. Then,

𝑑IF (|𝜓 ′ ⟩ , |𝜓 ∗ (𝑒 1 )⟩) = 1 − |⟨𝜓 ∗ (𝑒 1 )| 𝑃𝑒1 |𝜓 ′ ⟩| 2 = 1 − 𝜅|⟨𝜓 ∗ |𝜓(𝛽, 𝛼)⟩| 2 = (1 − 𝜅) + 𝜅𝑑IF (|𝜓(𝛽, 𝛼)⟩ , |𝜓 ∗ ⟩) ≤ (1 − 𝜅) + 𝜅𝐶 ′ ( 𝑝 ∗1 − 𝑝 1 (|𝜓(𝛽, 𝛼)⟩)) ≤ max{1/𝑝 ∗1 , 𝐶 ′ }( 𝑝 ∗1 (1 − 𝜅) + 𝜅( 𝑝 ∗1 − 𝑝 1 (|𝜓(𝛽, 𝛼)⟩))) ≲ 𝑝 ∗1 − 𝜅 𝑝 1 (|𝜓(𝛽, 𝛼)⟩)

(100)

where we use the single-mode stability bound in the second line. Manipulating via (5), 𝜅 𝑝 1 (|𝜓(𝛽, 𝛼)⟩) = 𝜅

∥𝑏 ′ ∥ 22 𝛽2 𝛽2 = = = 𝑝 1 (|𝜓 ′ ⟩) = 𝑝 1 (|𝜓⟩) 𝑍 (𝛽, 𝛼) 𝑍 (𝑏 ′ , 𝐴′ ) 𝑍 (𝑏 ′ , 𝐴′ )

(101)

Putting (99), (101), and (100) together, we have that 𝑑IF (|𝜓(𝑏, 𝐴)⟩ , |𝜓 ∗ (𝑥)⟩) ≲ 𝑝 ∗1 − 𝑝 1 (|𝜓(𝑏, 𝐴)⟩) for the parameter 𝑥 = 𝑏/∥𝑏∥ 2 , from which the claim trivially follows. □

77

B

Tensor representation of the truncated numerator

We prove Theorem 4.15. The main point is to represent the numerator by a tensor whose Frobenius norm is bounded independently of 𝑛. A bound on the values of 𝑝 alone would not give this coefficient bound. √ A linear map producing the truncated Gaussian vector. Set 𝑤(𝜃) = (1, 𝑏, 𝐴/ 2) ∈ X. We construct a linear map L 𝐾 : X ⊗𝐾 −→ 𝑃𝐾 F𝑛 , L 𝐾 𝑤(𝜃) ⊗𝐾 = 𝑃𝐾 |Φ(𝜃)⟩ , (102) √ with ∥L 𝐾 ∥ op ≤ 𝑐 𝐾 = (𝐾 + 1) 2 𝐾!. Since the creation operators commute, 𝑃𝐾 𝑒 𝐿𝑏 +𝑄 𝐴 |0⟩ =

∑︁ 𝑄 𝑟 𝐿 𝑠 𝐴 𝑏 2𝑟+𝑠≤𝐾

|0⟩ .

𝑟!𝑠!

We construct one linear map for each summand. The term indexed by (𝑟, 𝑠) uses 𝑟 copies of 𝐴 and 𝑠 copies of 𝑏, and lies in the (2𝑟 + 𝑠)-photon sector. The constant coordinate of 𝑤(𝜃) supplies the remaining 𝐾 − 𝑟 − 𝑠 input factors. We first record the normalization for passing from tensors to photon states. Let 𝜄𝑚 be the isometry from the 𝑚-photon sector onto Sym𝑚 𝐻 given by
√ 𝜄𝑚 𝑎ˆ † (𝑣 1 ) · · · 𝑎ˆ † (𝑣 𝑚 ) |0⟩ = 𝑚! Sym𝑚 (𝑣 1 ⊗ · · · ⊗ 𝑣 𝑚 ), (103) Í where 𝑎ˆ † (𝑣) = 𝑖 𝑣 𝑖 𝑎ˆ †𝑖 and Sym𝑚 is the orthogonal symmetrization projector. Indeed, for standard basis Î vectors 𝑣 𝑗 with occupation counts 𝑘 𝑖 , the creation √ vector has squared norm 𝑖 𝑘 𝑖 !, while the symmetrized Î tensor has squared norm 𝑖 𝑘 𝑖 !/𝑚!. The factor 𝑚! is thus the correct normalization. Fix 𝑟, 𝑠 ≥ 0 with 𝑚 = 2𝑟 + 𝑠 ≤ 𝐾. Let 𝐸𝑟 ,𝑠 : X ⊗𝐾 → 𝐻 ⊗𝑚 be the coordinate projection onto (Sym2 𝐻) ⊗𝑟 ⊗ 𝐻 ⊗𝑠 ⊗ C⊗ (𝐾 −𝑟 −𝑠) , √ followed by the canonical inclusion into 𝐻 ⊗𝑚 . Thus ∥𝐸𝑟 ,𝑠 ∥ op ≤ 1, and for 𝑤 = (𝑡, 𝑏, 𝐴/ 2), 𝐸𝑟 ,𝑠 𝑤 ⊗𝐾 = 𝑡 𝐾 −𝑟 −𝑠 2−𝑟/2 𝐴 ⊗𝑟 ⊗ 𝑏 ⊗𝑠 . Now, we are ready to define

√ L𝑟 ,𝑠 =

𝑚! −1 𝜄 Sym𝑚 𝐸𝑟 ,𝑠 , 𝑟/2 2 𝑟!𝑠! 𝑚

Í which corresponds to the (𝑟, 𝑠) term in the decomposition of Φ(𝜃). Using (103) and 𝑄 𝐴 = 12 𝑖, 𝑗 𝐴𝑖 𝑗 𝑎ˆ †𝑖 𝑎ˆ †𝑗 , we obtain √ 𝑡 𝐾 −𝑟 −𝑠 𝑚! −1 ⊗𝐾 L𝑟 ,𝑠 𝑤 = 𝜄𝑚 Sym𝑚 ( 𝐴 ⊗𝑟 ⊗ 𝑏 ⊗𝑠 ) 2𝑟 𝑟!𝑠! (104) 𝑡 𝐾 −𝑟 −𝑠 𝑟 𝑠 = 𝑄 𝐴 𝐿 𝑏 |0⟩ . 𝑟!𝑠! Í To conclude the construction, let L 𝐾 = 2𝑟+𝑠≤𝐾 L𝑟 ,𝑠 , viewing each output as an element of 𝑃𝐾 F𝑛 . Taking 𝑡 = 1 in (104) proves (102).

78

−1 is an isometry. Hence, on the full input space X ⊗𝐾 , Finally, 𝐸𝑟 ,𝑠 and Sym𝑚 are contractions and 𝜄𝑚

√︁ ∥L𝑟 ,𝑠 ∥ op ≤

(2𝑟 + 𝑠)! √ ≤ 𝐾!. 2𝑟/2𝑟!𝑠!

There are at most (𝐾 + 1) 2 admissible pairs (𝑟, 𝑠), so ∑︁ √ ∥L 𝐾 ∥ op ≤ ∥L𝑟 ,𝑠 ∥ op ≤ (𝐾 + 1) 2 𝐾!. 2𝑟+𝑠≤𝐾

The tensor coefficient bound.

∗ 𝜎 Define B𝐾 = L 𝐾 ˆ 𝐾 L 𝐾 . The support of 𝜎 ˆ 𝐾 and (102) imply

𝑝(𝜃) = ⟨𝑤(𝜃) ⊗𝐾 , B𝐾 𝑤(𝜃) ⊗𝐾 ⟩. Positivity gives B𝐾 ⪰ 0, while ∥ 𝜎 ˆ 𝐾 ∥ 𝐹 ≤ tr 𝜎 ˆ 𝐾 ≤ 1 gives ˆ 𝐾 ∥ 𝐹 ≤ (𝐾 + 1) 4 𝐾!. ∥B𝐾 ∥ 𝐹 ≤ ∥L 𝐾 ∥ 2op ∥ 𝜎 Set 𝐵 = 𝑅 2𝐾 ℎ B𝐾 and 𝑦 = 𝑤/𝑅 ℎ . These give (61) and ∥𝐵∥ 𝐹 ≤ Γ. Also ∥𝑤(𝜃) ∥ 22 = 1 + ∥𝜃 ∥ 2P ≤ 1 + 92 ℎ ≤ 𝑅 2ℎ ,

∥𝑤(𝜃) − 𝑤(𝜃 ′ ) ∥ 2 = ∥𝜃 − 𝜃 ′ ∥ P ,

which proves the stated properties of 𝑦. Lipschitz continuity and effective dimension. ∥𝑢 ⊗𝐾 − 𝑣 ⊗𝐾 ∥ 2 ≤ 𝐾 ∥𝑢 − 𝑣∥ 2 . Hence

For ∥𝑢∥ 2 , ∥𝑣∥ 2 ≤ 1, a telescoping argument shows that

|𝑔(𝑢) − 𝑔(𝑣)| ≤ ∥𝐵∥ op (∥𝑢 ⊗𝐾 ∥ 2 + ∥𝑣 ⊗𝐾 ∥ 2 ) ∥𝑢 ⊗𝐾 − 𝑣 ⊗𝐾 ∥ 2 ≤ 2𝐾Γ∥𝑢 − 𝑣∥ 2 . For the claim about “dimensionality reduction”, apply Lemma 6.5 of the extended version of [BBK+ 25]. Their lemma is quite general and will directly apply to our situation. The intuition is that a tensor 𝑇 of bounded Frobenius norm only has so many “strong” directions that can contribute to the injective norm, and they can be found by just flattening the tensor on each coordinate and taking the singular vectors from each SVD. We now apply it to our setting. Take the order-2𝐾 tensor with entries 𝑇𝛼,𝛽 = 𝐵 𝛼,𝛽 /Γ, where 𝛼, 𝛽 are length-𝐾 multi-indices. To plug into their lemma, ⟨𝑇, 𝑣¯ ⊗𝐾 ⊗ 𝑣 ⊗𝐾 ⟩ = 𝑔(𝑣)/Γ, and ∥𝑇 ∥ 𝐹 = ∥𝐵∥ 𝐹 /Γ ≤ 1. Set all lower-degree tensors to zero and use accuracy 𝜉/Γ. The cited lemma gives an efficiently computable complex subspace 𝑊 with dimC 𝑊 ≤ 8(𝐾 + 1) 6 (Γ/𝜉) 2 ,

|𝑔(𝑣) − 𝑔(𝑃𝑊 𝑣)| ≤ 𝜉

(∥𝑣∥ 2 ≤ 1).

Construction cost. We have dim X = 1 + 𝑛 + 𝑛(𝑛 + 1)/2 ≤ (𝑛 + 1) 2 and dim 𝑃𝐾 F𝑛 = 𝑛+𝐾 ≤ (𝑛 + 1) 𝐾 . 𝐾 Thus all matrices and tensors above have (𝑛 + 1) 𝑂 (𝐾 ) entries. The entries of L 𝐾 are computed using at most 𝐾! permutations for each of at most (𝐾 + 1) 2 terms. We use matrix multiplication to construct 𝐵, and taking SVDs (which are used in [BBK+ 25]) has the same (𝑛 + 1) 𝑂 (𝐾 ) cost. 79

C

Subgradient and projection oracles

Proof of Proposition 4.18. Subgradients. Here we describe how to compute the subgradient of 𝜆[∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 − 𝑡] + . We choose subgradient 0 when ∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 ≤ 𝑡, and otherwise take 𝜆 times the gradient of ∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 . The function 𝜃 ↦→ ∥𝑃𝑊 𝑦(𝜃) − 𝑧∥ 2 is 1/𝑅 ℎ -Lipschitz with respect to ∥ · ∥ P , by Theorem 4.15. Hence our choice of penalty subgradient has norm at most 𝜆/𝑅 ℎ . Adding ∇ P ℓ(𝜃), computed from (54), gives a subgradient 𝑔 𝑧 (𝜃) of 𝐽𝑧 satisfying ∥𝑔 𝑧 (𝜃) ∥ P ≤ 𝐺 ℎ + 𝜆/𝑅 ℎ . Clearly, all operations here are polynomial in 𝑛 and dimC 𝑊. Projection. The squared parameter distance is ∥𝑏 − 𝑏 0 ∥ 22 + 21 ∥ 𝐴 − 𝐴0 ∥ 2𝐹 , and the domain constraints √ separate between 𝑏 and 𝐴. Thus we project 𝑏 0√onto its Euclidean ball of radius 2 ℎ and minimize ∥ 𝐴 − 𝐴0 ∥ 2𝐹 over symmetric matrices satisfying ∥ 𝐴∥ 𝐹 ≤ ℎ and ∥ 𝐴∥ op ≤ 𝑟 ℎ . First, the matrix projection preserves the Takagi factor. Write 𝐴0 = 𝑈 diag(𝑠)𝑈 ⊤ , where 𝑠𝑖 ≥ 0. For any e = 𝑈 diag(𝑥)𝑈 ⊤ is still feasible matrix 𝐶, put 𝐷 = 𝑈 †𝐶𝑈 and 𝑥 𝑖 = max{Re 𝐷 𝑖𝑖 , 0}. The aligned matrix 𝐶 feasible, since ∑︁ e∥ 2 = e∥ op = max 𝑥 𝑖 ≤ max |𝐷 𝑖𝑖 | ≤ ∥𝐷 ∥ op = ∥𝐶 ∥ op . ∥𝐶 𝑥 𝑖2 ≤ ∥𝐷 ∥ 2𝐹 = ∥𝐶 ∥ 2𝐹 , ∥𝐶 𝐹 𝑖

𝑖

𝑖

Moreover, since 𝑠𝑖 ≥ 0, replacing 𝐷 𝑖𝑖 by 𝑥 𝑖 can only decrease its distance to 𝑠𝑖 . Unitary invariance of the Frobenius norm therefore gives ∑︁ ∑︁ ∑︁ 2 2 e∥ 2 = ∥ 𝐴0 − 𝐶 (𝑠 − 𝑥 ) ≤ |𝑠 − 𝐷 | + |𝐷 𝑖 𝑗 | 2 = ∥ 𝐴0 − 𝐶 ∥ 2𝐹 . 𝑖 𝑖 𝑖 𝑖𝑖 𝐹 𝑖

𝑖

𝑖≠ 𝑗

Thus an optimal matrix has the same Takagi factor 𝑈 as 𝐴0 . Second, project the singular-value vector. Since the Takagi factors are the same, it remains to solve the ordinary Euclidean projection problem n o √ 𝑠′ = argmin 12 ∥𝑥 − 𝑠∥ 22 : ∥𝑥∥ 2 ≤ ℎ, ∥𝑥∥ ∞ ≤ 𝑟 ℎ . 𝑥 ∈R𝑛

This projection onto the intersection of an ℓ2 ball and an ℓ∞ ball can be computed with polynomially many real-arithmetic operations. Since 𝑠 ≥ 0, its projection also satisfies 𝑠′ ≥ 0. The desired matrix projection is then 𝐴′ = 𝑈 diag(𝑠′ )𝑈 ⊤ . (105) □

80

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