arXiv CS open access

Quantum Federated Learning Based on Bures--Uhlmann Geometry for Heterogeneous Noisy Clients

2026 · arxiv_cs

arXiv CS · Papers · 2026

deep-learning machine-learning neural-networks

Conceptio Open Archive

Analyze & explore related research in Conceptio

Connect this record directly into your research workflow or AI agent. Explore citations across open repositories, query full text, and export structured proof bundles.

Explore in Search →Connect via MCP / API ↗

Quantum Federated Learning Based on Bures–Uhlmann Geometry for Heterogeneous Noisy Clients Haruki Emori ,1, 2, ∗ Masaki Uchihara ,3 and Yuuki Tokunaga

arXiv:2608.28379v1 [quant-ph] 28 Aug 2026

Graduate School of Information Science and Technology, Hokkaido University, Kita 14, Nishi 9, Kita-ku, Sapporo, Hokkaido 060-0814, Japan 2 RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS), RIKEN, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan 3 Department of Diabetes, Endocrinology, and Metabolism, National Center for Global Health and Medicine, Japan Institute for Health Security, 1-21-1 Toyama, Shinjuku-ku, Tokyo 162-8655, Japan (Dated: August 31, 2026) Quantum federated learning enables collaborative model training across quantum devices without sharing raw data, and it faces the data and hardware heterogeneity inherent to noisy quantum devices. Utilizing the quantum geometric tensor is a natural remedy, yet pure-state approaches and diagonal approximations discard the correlations that encode parameter incompatibility. To address this, we extend the parameter-space geometry to the mixed states that noisy clients actually prepare. The real part of the resulting mixed-state geometric tensor is the Bures metric, which measures how fast the physical state changes under parameter variation, and the imaginary part is the mean Uhlmann curvature, which quantifies the incompatibility of estimating multiple parameters simultaneously. Accordingly, we employ the Bures metric as a local preconditioner and use the mean Uhlmann curvature to develop an achievable-precision aggregation rule that dynamically downweights unreliable clients. Furthermore, we establish theoretical guarantees by proving a convergence theorem and a variance-dominance proposition. Empirical evaluations on a trapped-ion quantum emulator demonstrate that the proposed method maintains high accuracy across diverse deviceheterogeneity conditions and outperforms standard federated averaging, whose accuracy degrades under strong noise.

INTRODUCTION

Machine learning increasingly relies on data that are private, regulated, or too large to centralize, such as clinical records held by separate hospitals, sensor logs on edge devices, or financial transactions inside different institutions. Federated learning (FL) was introduced to train a shared model on such siloed data without moving the raw data off the owning device, in which each client optimizes a local model and shares only parameter updates with a coordinating server [1]. This addresses societal needs for data privacy and regulatory compliance, and it inherits two structural difficulties. First, client datasets are non-independent and non-identically distributed (non-IID), which slows or biases the global model. Second, communication of model updates is costly and is itself a privacy attack surface. Quantum federated learning (QFL) augments this picture with quantum resources. Clients hold variational quantum circuits (VQCs) [2–4], encode their classical data into quantum states, and train by measuring expectation values, and the server aggregates the circuit parameters [5–7]. The motivation is twofold. On the modeling side, VQCs access high-dimensional Hilbert spaces and may represent correlations that are hard classically. On the systems side, quantum communication primitives such as quantum key distribution can protect the shared updates with physicallayer security [6]. A comprehensive account of QFL architectures, communication schemes, optimization strategies, security mechanisms, and applications is given in the recent survey of Nguyen et al. [6]. Prior QFL work has, for example, introduced Fisher-information–based aggregation for non-IID data (QFedFisher) [8], adaptive two-gain aggregation over noisy quantum channels [9], layerwise training for hardware-heterogeneous clients [10], and the use of intrinsic hardware noise for differential privacy [11]. A difficulty specific to QFL, and largely absent from classical FL, is device heterogeneity. In the noisy intermediatescale quantum (NISQ) era each client runs on hardware with a different gate-error rate, coherence time, and number of measurement shots. Two clients with identical data and different devices produce updates of different reliability, and naive averaging is influenced most by the worst devices. Our numerical experiments make this concrete. When the unreliable clients have depolarizing rates of 0.4–0.6, federated averaging of the same ansatz reaches a roundaveraged accuracy of 0.77–0.78 on a ten-class digit task with a worst evaluated round as low as 0.67, whereas the geometry-aware rule proposed here reaches 0.81–0.82 with a worst round above 0.73. ∗ emori.haruki.i8@elms.hokudai.ac.jp

2 A geometry-aware treatment of the parameter space is a natural remedy. The relevant geometric object is the quantum geometric tensor (QGT) Qij = gij − 2i ωij [12–15], whose real part gij is the Fubini–Study metric and whose imaginary part ωij is the Berry curvature. A subtlety here motivates the present work. For a pure state the real part of the QGT equals one quarter of the quantum Fisher information matrix (QFIM), FijQ = 4gij [16, 17]. Any optimizer that preconditions the gradient with the real part of the pure-state QGT is, by definition, the quantum natural gradient (QNG) of Stokes et al. [16], and transplanting it into a federated loop adds no geometric novelty on the local-optimization side. The diagonal approximation commonly used to reduce the measurement cost of the metric discards the off-diagonal correlations that, as we show below, carry the quantum incompatibility information once the state is mixed. These two points limit any pure-state QGT approach to QFL. We resolve both points by moving the geometry to the regime in which QFL clients actually operate. Because of hardware noise, a client prepares a mixed state ρk (θ) rather than a pure state |ψ(θ)⟩. The geometry for a density matrix is the mixed-state QGT, whose real part is the Bures metric, the minimal monotone metric on density matrices [18–21], and whose imaginary part is the mean Uhlmann curvature [19, 21]. The Bures metric reduces to the Fubini–Study metric for pure states and to the classical Fisher matrix for commuting families, and it differs from both in the intermediate, O(1)-noise regime where QFL operates. This is the open direction that Stokes et al. flagged in their discussion as the candidate for noisy devices [16], and we develop it in the QFL context. a. Contributions. (i) We identify the pure-state equivalence QGT(real)=QFIM=QNG, and we extend QFL beyond it to the mixed-state Bures–Uhlmann geometry. (ii) We make the roles of the two parts precise. The symmetric (Bures) part is a reparametrization-invariant local preconditioner, and the antisymmetric (Uhlmann) part enters, through the quantum Cramér–Rao bound [22–25], an achievable-precision aggregation rule. (iii) We replace the diagonal approximation by an exact block-diagonal estimator with O(L) overhead and bound its truncation error. (iv) We derive the closed-form depolarizing attenuation κ(p) and show that for non-global noise the correction is matrix-valued, and the full mixed-state geometry is needed. (v) We prove a convergence theorem, and we prove in addition (Proposition 1) that the proposed aggregation never has a larger residual variance than federated averaging, with equality if and only if the devices are homogeneous. The reduction of the aggregated variance is therefore a consequence of the analysis rather than a tuning artefact. (vi) We evaluate on the full MNIST database (ten-class task, digits 0–9), using the trapped-ion emulator reimei-E of the reimei quantum computer [26] through Quantinuum Nexus under four device-heterogeneity conditions, averaging over three independently seeded federations. The proposed method attains the highest round-averaged accuracy in every condition, and the size of its advantage over federated averaging is largest where the closed-form weight ratio predicts the strongest down-weighting and narrows toward parity where the devices are reliable. The approach points toward robust, privacy-preserving collaborative learning on the heterogeneous quantum hardware of the emerging quantum internet.

II. A.

PRELIMINARIES

Quantum federated learning

We first describe QFL quantitatively for readers familiar with quantum mechanics and not with FL. A qubit state is a unit vector |ψ⟩ = α|0⟩ + β|1⟩ ∈ C2 with |α|2 + |β|2 = 1, and an n-qubit register lives in H = (C2 )⊗n of dimension N = 2n . A variational quantum classifier maps a classical input x ∈ Rdin to a parametrized state |ψ(x; θ)⟩ = U (θ) UEnc (x)|0⟩⊗n ,

(1)

where UEnc (x) encodes the data by angle encoding, Ry (xj ) = e−ixj Y /2 , and U (θ) is a trainable unitary with real parameters θ ∈ Rd . The model output is built from the expectation values of a fixed set of observables {Pf }, ϕf (x; θ) = Tr Pf |ψ(x; θ)⟩⟨ψ(x; θ)| ,

(2)

read out by repeated measurement, with the set specified in Sec. III C. Gradients ∂i ϕf are obtained on hardware by the parameter-shift rule [27, 28]. In FL there are K clients. Client k owns a private dataset Dk of size nk that never leaves the device, and FK P D = k=1 Dk with |D| = k nk . With a per-sample loss ℓ, here the softmax cross-entropy, the local objective is fk (θ) =

1 nk

X (x,y)∈Dk

ℓ y, ŷ(x; θ) ,

(3)

3 and the federated goal is to minimize the global objective F (θ) =

K X

pk fk (θ),

k=1

pk =

nk . |D|

(4)

Training proceeds in synchronous rounds. The server broadcasts the global parameters θ (t) , each client performs (t+1) local optimization on fk and returns its updated parameters θk , and the server aggregates them into θ (t+1) . The P (t+1) canonical rule, federated averaging (QFedAvg), uses the data-size weights, θ (t+1) = k pk θk . QFedFisher [8] replaces pk by weights built from a diagonal classical Fisher information. The central question of this paper is what aggregation weights are geometrically and statistically appropriate when clients are noisy and heterogeneous, and how the parameter-space geometry should shape the local update. B.

Pure-state quantum geometric tensor

We recall the QGT in a form that makes its meaning transparent, following the geometric formulation of quantum mechanics [12, 13]. Consider a smooth family of pure states |ψ(θ)⟩. The naive distance obtained from ⟨∂i ψ|∂j ψ⟩ is not gauge invariant, because a local phase change |ψ⟩ → eiα(θ) |ψ⟩, which leaves all physical observables unchanged, alters it. Removing the phase (Berry) connection Ai = i⟨ψ|∂i ψ⟩ ∈ R yields the gauge-invariant quantum geometric tensor on the projective space of rays [12], QPure (θ) = ⟨∂i ψ|∂j ψ⟩ − ⟨∂i ψ|ψ⟩⟨ψ|∂j ψ⟩ = gij − 2i ωij . ij

(5)

Because the Hilbert-space inner product is Hermitian, the real part gij = Re QPure is symmetric and the imaginary ij Pure part ωij = −2 Im Qij is antisymmetric. The real part is the Fubini–Study metric [12], and it measures the rate at which the physical state changes under parameter variation, because the fidelity obeys [12] X |⟨ψ(θ)|ψ(θ + dθ)⟩|2 = 1 − gij dθi dθj + O(dθ3 ). (6) ij

The antisymmetric part ωij is the Berry curvature, the flux of the emergent gauge field Ai associated with the geometric (Berry) phase [29]. Thus g describes how far the state moves, while ω describes the phase holonomy accumulated, and one needs g to endow the parameter manifold with a notion of length and steepest descent. For learning, the relevance of g is twofold. It is the unique unitarily invariant metric on pure states, and it provides a parametrization-independent notion of the natural descent direction, and it equals one quarter of the pure-state quantum Fisher information matrix, FijQ = 4gij [17], which links geometry to estimation precision. The quantum natural gradient (QNG) [16, 30] uses this by preconditioning the Euclidean gradient with g, −1 θ (t+1) = θ (t) − η g(θ (t) ) + ε11 ∇L(θ (t) ), (7) where L(θ) is the differentiable training objective minimized locally, in the federated setting the client empirical risk fk (θ) of Eq. (3), and ∇L(θ (t) ) is its Euclidean gradient at the current iterate θ (t) . Here η > 0 is the learning rate and ε > 0 is a Tikhonov regularizer that guarantees the inverse (g + ε11)−1 exists even when g is rank deficient. Equation (7) corrects for the curvature of the state manifold and mitigates barren plateaus [31, 32]. The observation for QFL is that Eq. (7) is QNG, and any method that preconditions with the real part of the pure-state QGT reproduces it exactly. A pure-state QGT-based QFL therefore cannot claim novelty at the level of the local update unless the geometry itself is changed. This is what the mixed-state extension below accomplishes. C.

Bures metric and mean Uhlmann curvature for mixed states

On real hardware client k realizes a mixed state produced by a noise channel Nk rather than the pure state (1), ρk (θ) = Nk U (θ)|0⟩⟨0|U † (θ) , Tr ρ2k < 1. (8) The geometric object appropriate to a density matrix is built from P the symmetric logarithmic derivative (SLD) Li , defined by ∂i ρk = 12 (Li ρk + ρk Li ) [22–25]. In the eigenbasis ρk = n λn |n⟩⟨n| the mixed-state QGT is Bur 1 i Uhl QMix (9) k,ij := 4 Tr ρk Li Lj = gk,ij − 2 ωk,ij ,

4 whose real (Bures) and imaginary (mean Uhlmann curvature) parts have the explicit forms [21] 1 X Re ⟨m|∂i ρk |n⟩⟨n|∂j ρk |m⟩ Bur 1 Q,SLD = , gk,ij = 4 Fk,ij 2 λm + λn λm +λn >0 X (λm − λn ) Im ⟨m|∂i ρk |n⟩⟨n|∂j ρk |m⟩ Uhl i . ωk,ij = 2 Tr ρk [Li , Lj ] = (λm + λn )2

(10)

(11)

λm +λn >0

Equations (10)–(11) are the mixed-state generalizations of the real and imaginary parts of (5). The difference from the pure case is conceptual and quantitative, and it is summarized by three points. (a) Distinctness from QFIM and classical Fisher. By Petz’s theorem [20], monotone metrics on density matrices are not unique, and the Bures metric is the minimal one. It reduces to the Fubini–Study metric in the pure limit ρk → |ψ⟩⟨ψ| and to one quarter of the classical Fisher matrix in the commuting limit [ρk , ∂i ρk ] = 0, and for an O(1)-noise state it coincides with neither [21]. QFL clients operate in this intermediate regime, and a natural choice of metric is gkBur rather than the pure-state QGT used by QNG or the classical Fisher matrix used by QFedFisher [8]. (b) Quantum content in the imaginary part. The mean Uhlmann curvature (11) vanishes for pure states and for commuting families and is nonzero whenever [ρk , ∂i ρk ] ̸= 0. It measures the incompatibility of estimating different parameters simultaneously and has no classical analogue [21]. Discarding the off-diagonal elements by a diagonal approximation destroys this quantum information, which we avoid in Sec. III. (c) Separate roles of the two parts. For a real parameter update the descent direction solves g δθ = −η∇L, and any antisymmetric matrix ω contributes nothing because δθ⊤ ω δθ = 0. The imaginary part cannot, and must not, be inserted into the gradient step, which is why QNG uses only the real part. The imaginary part has a distinct statistical role. When the SLDs do not commute, the attainable covariance of a multiparameter estimate exceeds the SLD quantum Cramér–Rao bound (the Holevo bound), and the excess is governed by ωkUhl [21, 25, 33, 34]. We therefore use the Bures part as the local preconditioner and the Uhlmann part to quantify the reliability of each client’s reported update (Sec. III B). This division of labour uses both parts of the QGT and does not insert the imaginary part into the gradient step.

III.

PROPOSED METHOD QFEDQGT

The proposed scheme has two ingredients, a mixed-state natural-gradient local update that uses the Bures metric, and a server aggregation rule whose weights are the achievable precision of each client. Both are estimated on hardware by an exact block-diagonal procedure, and the diagonal approximation is never used. We first fix notation. The noiseless (pure) QGT real part of the shared ansatz |ψ(θ)⟩ = U (θ)|0⟩⊗n is h i Pure G (θ) ij = Re ⟨∂i ψ|∂j ψ⟩ − ⟨∂i ψ|ψ⟩⟨ψ|∂j ψ⟩ , (12) which is a real symmetric positive-semidefinite dq × dq matrix on the quantum parameters θq (Sec. III C). Client k’s Bures metric Gk = gkBur from (10) reduces to a noise attenuation of (12) for the noise model treated below. A.

Local update and its reparametrization invariance

e k = ∇fk + ξk Client k estimates the local gradient by the parameter-shift rule, obtaining an unbiased estimate ∇f (t) Bur (t) with E[ξk ] = 0 and Cov(ξk ) = Vk , and the block-diagonal Bures metric Gk = gk (θk ). The local step is the mixed-state natural gradient (t+1)

θk

(t) b (t) + ε11 −1 ∇f e k (θ (t) ), = θk − ηLoc G k k

(13)

b k = Gk /ḡk is normalized by its mean diagonal ḡk = 1 Tr Gk to make the step size scale invariant, and ε > 0 where G dq regularizes the inverse. This update is invariant, to first order, under a smooth invertible reparametrization ϕ = Φ(θ) with Jacobian J. The metric transforms as a covariant 2-tensor, g (ϕ) = J −⊤ g (θ) J −1 , and the gradient as a covector, ∇ϕ L = J −⊤ ∇θ L, hence (g (ϕ) )−1 ∇ϕ L = J (g (θ) )−1 ∇θ L. The update direction is the same tangent vector pushed forward by J and does not depend on the coordinate chart. The Euclidean gradient lacks this property, and the invariance is what allows clients that use different circuit parametrizations or encodings to be combined consistently.

5 B.

Achievable-precision aggregation

We now define what is aggregated and with what weights. Let the achievable precision of client k be the inverse e covariance of the quantity the server combines, the local update direction dk = G−1 k ∇fk . Because dk is linear in the gradient noise, its fluctuation covariance is −1 Σk = Cov(dk ) = G−1 k Vk Gk ,

(14)

and the metric appears as a noise-shaping Jacobian. We define the precision matrix −1 Λk := Σ−1 k = Gk Vk Gk .

(15)

The covariance Vk is not arbitrary. It is lower-bounded by the quantum Cramér–Rao bound for Mk measurement shots, together with the incompatibility correction that the mean Uhlmann curvature induces, i 1 h Q,SLD −1 (16) + C ωkUhl . Fk Vk ≥ Mk Here C(·) ≥ 0 is the incompatibility penalty. Writing the antisymmetric mean Uhlmann curvature as the matrix Uhl [ Uk ]ij = ωk,ij and abbreviating Fk ≡ FkQ,SLD , an explicit positive-semidefinite representative obtained from the gap between the Holevo and SLD bounds is −1 −1 −1 −1 C ωkUhl = Fk−1 Uk Fk−1 U⊤ (17) k Fk = − Fk Uk Fk Uk Fk ≥ 0, Uhl where the last equality uses U⊤ k = −Uk . By construction C(ωk ) vanishes if and only if the parameters are compatible (Uk = 0, for example for pure states), and it grows with the noncommutativity of the SLDs. It quantifies how much harder it is to estimate all parameters of a noisy client at once. b = P Ak dk with P Ak = 11, which keeps the combination unbiased The server combines the directions linearly, d k k P ⊤ for the consensus direction, and it chooses the matrices Ak to minimize the aggregated variance Tr k A k Σk A k . By Lagrange multipliers this generalized-least-squares (maximum-ratio-combining) problem has the solution

Aopt = k

j Λj

−1

Λk ,

min Tr

h P −1 i ⊤ A Σ A = Tr Λ , k k k k k k

(18)

and each client is weighted by its precision and the residual variance is the inverse of the total precision [35, 36]. The server then updates the global parameters on the tangent space, the first-order Bures barycenter, X opt (t+1) θ (t+1) = θ (t) + ηSrv A k θk − θ (t) , (19) k

with a server step ηSrv ∈ (0, 1] that damps client drift. A consequence of (18) is that the aggregated variance of the proposed rule never exceeds that of QFedAvg. Proposition 1 (Variance dominance over QFedAvg). P P ⊤ Let A = { {Ak } : k Ak = 11 } and let V({Ak }) = Tr k Ak Σk Ak . Then for every admissible weighting, and in particular for the QFedAvg choice Ak = pk 11, h P −1 i P Tr = minA V ≤ V {pk 11} = k p2k Tr Σk , (20) k Λk P with equality if and only if Λk = pk j Λj for all k, that is, if and only if the clients’ precisions are already proportional to their data shares, in particular if all devices are identical and pk = 1/K. Proof. The set A is affine and the map {Ak } 7→ V is a sum of the strictly convex quadratics Tr(Ak Σk A⊤kP ) when every Σk ≻ 0, and a minimizer therefore k pk 11 = 11, P 2exists and is unique. The choice Ak = pk 11 is admissible because P hence minA V ≤ V({pk 11}) = k pk Tr Σk . The minimizer is computed in Eq. (18) and gives value Tr[( k Λk )−1 ], which establishes the equality on the left of (20). Uniqueness of the minimizer makes the inequality tight exactly P P when pk 11 = Aopt = ( j Λj )−1 Λk for all k, that is, Λk = pk j Λj . k Proposition 1 concerns the aggregated variance alone and does not by itself assert an advantage in any other figure of merit. Under device homogeneity the proposed rule degenerates to QFedAvg, and under any heterogeneity it reduces the aggregated variance, which by Theorem 1 is the quantity that sets the asymptotic error floor.

6 For a scalable implementation we use the data-size × precision scalar weights p k ϱk , wk = P j pj ϱj

ϱk =

κ(pk )2 ∥GPure ∥2F , vk 1 + γ rk

rk =

∥Uk ∥2F , ∥Gk ∥2F

(21)

where κ(pk ) is the depolarizing attenuation of Sec. III E, vk is the per-component shot-noise variance of the gradient estimate, rk is the relative Uhlmann incompatibility, and γ ≥ 0 tempers the down-weighting to keep noisy clients discounted and not discarded. Equation (21) equals ϱk = Tr Λk /(1 + γrk ) under the isotropic model Vk ≃ vk 11 of Sec. III E, and it reduces to QFedAvg when all devices are identical, while increasingly favouring high-shot, low-noise, compatible clients as heterogeneity grows. To keep the rule a discount rather than a veto we retain a floor λfl , wk ← (1 − λfl )wk + λfl pk , which guarantees every client a nonzero weight, and no data are silently discarded. The floor is adapted to the heterogeneity of the client pool through λfl = λ0 + (λmax − λ0 ) s,

p − p̄ hi s = clip , 0, 1 , phi − plo

p̄ =

pk pdep k ,

(22)

where pdep is the depolarizing rate of client k and p̄ is the data-weighted mean depolarizing rate of the pool. When the k pool is uniformly reliable (p̄ ≤ plo ) the floor takes its upper value λmax and the rule approaches data-size averaging, consistent with the homogeneous limit of Proposition 1, and when the pool is strongly heterogeneous (p̄ ≥ phi ) the floor takes its lower value λ0 and the precision weighting acts in full.

Shared model with quantum feature map and Pauli-expectation readout

All three schemes share the same model, and any performance difference is attributable to the geometry and the aggregation alone. The model is a dressed variational classifier [37], a quantum feature map followed by a trained classical readout. The feature map is the circuit of Fig. 1, with L layers, each of which re-uploads the data through Ry rotations [38], applies a trainable Rz Ry Rz block per qubit, and closes with a linear CNOT entangler. The quantum parameters are θq ∈ Rdq with dq = 3nL. The measured features are the expectation values of a compact set of Pauli observables, {Pf } = {Zq }nq=1 ∪ {Xq }nq=1 ∪ {Yq }nq=1 ∪ {Zq Zr }q<r ,

, Mf = 3n + n(n−1) 2

(23)

which for n = 6 gives Mf = 33. This is the multiclass generalization of a Pauli-Z readout. Each ϕf = ⟨Pf ⟩ ∈ [−1, 1] is an O(1) quantity, unlike the 2n computational-basis probabilities, which vanish as 2−n and make the readout illconditioned at larger n, and the whole set is obtained on hardware from only three measurement settings. The Z setting supplies all Zq and Zq Zr , and the X and Y settings supply Xq and Yq . The class scores are produced by a trainable readout head acting on ϕ = (ϕ1 , . . . , ϕMf ), z(x) = W2 tanh W1 ϕ(x; θq ) + b1 + b2 , ŷ = arg max zc , (24) c

with θh = (W1 , b1 , W2 , b2 ) and H hidden units. The full parameter vector is θ = (θq , θh ). A linear head (H = 0) is the literal Pauli-expectation readout, and it is capped by the linear separability of the PCA features and cannot exceed about 0.85 on the ten-class digit task. The single tanh layer removes that ceiling while keeping the readout shared and keeping the quantum block small. The Bures geometry acts only on the quantum block θq , the head is a classical Euclidean object, and the mechanism of Eqs. (13)–(21) is unaffected and the head is identical for QFedAvg, QFedFisher and QFedQGT.

Hardware estimation without diagonal truncation

Bur The diagonal approximation gkBur ≈ diag(gk,ii ) deletes the off-diagonal elements, which by (11) are where the incompatibility Uk lives, and it would disable the mechanism of Eqs. (17)–(21). We do not use it. Instead we exploit the layered structure of the ansatz U (θ) = VL (θL )WL · · · V1 (θ1 )W1 . For the pure-state QGT (l) the block belonging to layer l is Gij = ⟨ψl |Ki Kj |ψl ⟩ − ⟨ψl |Ki |ψl ⟩⟨ψl |Kj |ψl ⟩, with generators Ki and the sub-circuit

P |0⟩

Ry (x1 )

Rz P

|0⟩

Ry (x2 )

Rz P

|0⟩

Ry (x3 ) encode

trainable Vl (θl )

entangle Wl

FIG. 1. One layer of the shared variational quantum classifier, drawn for three qubits, with the experiments using n = 6 and L = 3. Each layer re-uploads the data through Ry (xj ) (green), applies a trainable single-qubit block Rz Ry Rz (blue), and a linear CNOT entangler (pink). After L layers the Pauli observables (23) are measured in three settings (Z, X, Y ) and fed to the shared readout head (24). The three QFL methods use this same circuit and the same head, and differ only in the local update and the aggregation.

state |ψl ⟩ prepared just before layer l, and one circuit per layer suffices [16]. The Bures metric is the Hessian of the p √ √ 2 ρσ ρ , Uhlmann fidelity FB (ρ, σ) = Tr Bur (25) gk,ij = − 12 ∂θi′ ∂θj′ FB ρk (θ), ρk (θ ′ ) θ′ =θ , and we estimate each layer block, including its off-diagonal elements, from parameter-shifted fidelities [39–41]. This retains all intra-layer correlations at O(L) measurement cost rather than the O(d2q ) of the full matrix, and the error from neglecting inter-layer blocks is bounded in Appendix B. E.

Exactly solvable case of global depolarizing noise

For the global depolarizing model ρk = (1 − pk )|ψ⟩⟨ψ| + pk 11/N with N = 2n , Eq. (10) can be evaluated in closed form (Appendix A), Q,SLD Fk,ij = κ(pk ) FijQ,Pure ,

κ(p) =

(1 − p)2 , 1 − p + 2p/N

(26)

and Gk = κ(pk ) GPure . One has κ → 1 as p → 0, recovering the pure-state QFIM, and κ ≈ 1 − p for N ≫ 1. In this model the correction is a scalar. The natural-gradient direction is unchanged, and the achievable precision is attenuated, which is the physical reason a noisy client should be down-weighted. With Vk ≈ vk 11 and vk =

1 Mk (1 − pk )2

(27)

(finite-shot variance amplified by the depolarizing attenuation of the measured expectation values, ⟨P ⟩ → (1−pk )⟨P ⟩), the precision (15) becomes Λk = κ(pk )2 vk−1 (GPure )2 , and Tr Λk = κ(pk )2 ∥GPure ∥2F Mk (1 − pk )2 ,

(28)

that is, a noisy client is suppressed by the product κ(pk )2 (1 − pk )2 and by its shot budget Mk . The two factors are asymmetric. The precision is only linear in the shot budget Mk and quartic in (1 − pk ), through κ2 (1 − p)2 ≃ (1 − p)4 for N ≫ 1, and the depolarizing rate rather than the shot scarcity is the dominant determinant of a client’s reliability. Section V confirms this prediction, in that the accuracy gap between QFedQGT and QFedAvg grows with pk and depends only weakly on Mk . For local (per-qubit) depolarizing or coherent errors the correction is matrix-valued and Uk ̸= 0, and the full mixedstate geometry is required, and for the coherent miscalibration used in our experiments we take the leading-order proxy rk ≃ p2k (Appendix D).

8 Algorithm: QFedQGT 1: Input: clients {1, . . . , K}, rounds T , server step ηSrv , local step ηLoc , temper γ, floor λfl (0)

(0)

2: initialize global parameters θ (0) = (θq , θh ) 3: for t = 0, . . . , T − 1 do 4: server broadcasts θ (t) and the block-diagonal GPure (θ (t) ) 5: for each client k in parallel do 6: θk ← θ (t) 7: for local step = 1, . . . , τ do e k (parameter shift) and Gk = κ(pk )GPure 8: estimate ∇f

b k + ε11)−1 ∇ e q fk , θh,k ← θh,k − ηLoc ∇ e h fk 9: θq,k ← θq,k − ηLoc (G 10: end for 11: estimate κ(pk ), vk , rk and form Tr Λk and wk 12: send (θk , wk ) to server 13: end for P 14: θ (t+1) ← θ (t) + ηSrv k wk (θk − θ (t) ) 15: end for 16: return θ (T )

▷ Eqs. (12)–(10)

▷ Eqs. (15)–(21)

Protocol

Algorithm summarizes the procedure. Each client performs a fixed number τ of local minibatch steps per round, and the local work per communication round is a controlled quantity. All client-side randomness is seeded deterministically from the pair (round, client), and a run is reproducible. The three compared methods share the VQC of Fig. 1 and the readout (24), and differ only in the local update and the aggregation weights. QFedAvg uses wk = pk and a plain (Adam) local optimizer, QFedFisher uses layerwise diagonal classical Fisher weights, and QFedQGT uses Eqs. (13) and (21).

IV.

CONVERGENCE ANALYSIS

We analyze one local step per round, and the multi-step case follows by treating the accumulated update as a stochastic direction. We state the assumptions and explain why each is needed. Assumption 1. (A1) (L-smoothness) F is L-smooth, that is, ∇F is L-Lipschitz, ∥∇F (u) − ∇F (v)∥ ≤ L ∥u − v∥, equivalently F (v) ≤ 2 F (u) + ∇F (u)⊤ (v − u) + L2 ∥v − u∥ . This bounds how fast the gradient varies and lets a finite step decrease the 2 loss. (A2) (Polyak–Lojasiewicz) F satisfies ∥∇F (θ)∥ ≥ 2µ (F (θ) − F ∗ ) with µ > 0, where F ∗ = minθ F (θ). This holds for µ-strongly convex F and, locally, near a non-degenerate minimum, and it converts gradient norm into objective suboptimality. (A3) (unbiased bounded noise) the gradient noises ξk are unbiased (E[ξk ] = 0), have P 2 1 2 covariance Vk , and are independent across clients. (A4) (bounded heterogeneity) K k ∥∇fk (θ) − ∇F (θ)∥ ≤ ζ , where ζ ≥ 0 measures the non-IID spread of the client P gradients and is 0 in the IID case. (A5) (metric regularity) λmin 11 ≤ Gk ≤ λmax 11 with 0 < λmin ≤ λmax , and k ∥Ak ∥ ≤ β. This keeps the preconditioner well conditioned, away from barren-plateau degeneracy, and the weights bounded. (A6) (descent alignment) there exists c > 0 with P 2 −1 ∇F⊤ A G ∇f ≥ c ∥∇F ∥ , which states that the aggregated natural-gradient direction is a descent direction, k k k k holding under (A4)–(A5) when the heterogeneity is not dominant. Plausibility of the assumptions. Each hypothesis is standard in the federated-optimization or the variational-quantum literature, and is satisfied in the regime studied here. (A1) holds for VQC objectives because each circuit output (2) is a finite trigonometric polynomial in θ with frequencies fixed by the generators of U (θ) [4, 42], hence C ∞ with uniformly bounded second derivatives, and ∇F is Lipschitz with an L set by the number of layers and the spectral norm of the generators. L-smoothness is also the baseline assumption in non-convex stochastic-optimization analyses [43]. (A2) is the Polyak–Lojasiewicz condition, weaker than strong convexity, and is the standard relaxation under which linear convergence is proved for over-parametrized and non-convex models [44]. For VQCs it holds locally in the basin of a non-degenerate minimum, once training has escaped the initialization plateau, and it underlies convergence guarantees for FedProx and related FL methods [36]. (A3) is exact for the parameter-shift estimator, in which shifting a parameter by ±π/2 yields an unbiased gradient component whose only stochasticity is finite-shot multinomial sampling, hence zero mean and a finite, shot-determined

9 covariance Vk [27, 28], with independence across clients following from independent hardware executions. (A4) is the canonical bounded-gradient-dissimilarity hypothesis used to quantify non-IID heterogeneity in federated optimization [36, 45], with ζ = 0 recovering the IID case, and our Dirichlet-α partition controls ζ through α. (A5) requires the preconditioner to stay well conditioned. The Tikhonov shift ε11 in (13) enforces λmin ≥ ε > 0 by b k = Gk /ḡk bounds λmax , which is the mechanism by which natural-gradient construction, and the normalization G P methods avoid the vanishing curvature of barren plateaus [3, 31], and k ∥Ak ∥ ≤ β follows from the normalized weights (21) and the floor λfl . (A6) states that the precision-weighted aggregate is, on average, a descent direction. Under (A4)–(A5) the aggregate differs from the true gradient by a term controlled by ζ and λmin , and a strictly positive alignment constant c exists whenever the heterogeneity does not dominate the curvature, a condition analogous to the bounded-dissimilarity requirements under which FedProx and FedNova converge [36, 45]. Theorem 1 (Robust convergence of QFedQGT). Let Assumption 1 hold and let the server step satisfy η ≡ ηSrv ≤ η0 with η0 = min 1/L, cλ2min /(2Lβ 2 ) . Then the global iterate θT produced by the precision-weighted rule (18) satisfies P Lη 2 ζeff + Tr ( k Λk )−1 , E[F (θT ) − F ∗ ] ≤ (1 − cηµ)T F (θ0 ) − F ∗ + 2cµ

(29)

where F ∗ = min F , T is the number of rounds, L and µ are the constants of (A1)–(A2), c is the alignment constant 2 of (A6), Λk are the precision matrices (15), and ζeff = 2β 2 K ζ 2 /λ2min is the effective heterogeneity floor built from ζ (A4) and β, λmin (A5). Equation (29) separates into the two effects that the experiments below exhibit. The first term decays geometrically at rate (1 − cηµ), and the number of communication rounds needed to reach a target suboptimality ϵ is Tϵ ≃ log(1/ϵ)/(cηµ). Anything that raises the alignment constant c or admits a larger stable step η reduces the round count. The Bures preconditioner contributes to both. It equalizes the curvature, which reduces λmax /λmin and with it the effective condition number, and through η0 = min{1/L, cλ2min /(2Lβ 2 )} this enlarges the admissible step, and the precision weights raise c by removing the mis-aligned contributions of the unreliable clients from the aggregate. P The second term is the residual error floor. Its variance contribution Tr[(P k Λk )−1 ] is the minimal aggregated variance (18), which P by Proposition 1 is never larger than the QFedAvg value k p2k Tr Σk , and it shrinks as the total achievable precision k Λk grows. A self-contained proof is given in Appendix C. V.

NUMERICAL EXPERIMENTS A.

Setup

We compare the three methods on the full MNIST database of handwritten digits [46], restricted to the ten classes {0, 1, . . . , 9} (“MNIST-10”), with 3000 training images per class (30,000 in total) drawn from the official training split. The ten-class task is a stringent multiclass setting. A random classifier scores 1/10 = 0.10, a linear classifier on the encoded features saturates below 0.9, and the task therefore discriminates between aggregation rules. The shared model is the dressed classifier of Sec. III C with n = 6 qubits and L = 3 layers, hence dq = 3nL = 54 quantum parameters, Mf = 3n + n(n − 1)/2 = 33 measured Pauli observables, and a tanh readout head with H = 256 hidden units. The 784 raw pixels are reduced by principal-component analysis to nf = nL = 18 components, standardized, and scaled to small re-uploading angles, with the PCA basis and the scaler fitted on the training split only, and no test information leaks into training. The depth was selected by a sweep, and on the federated loop the achieved accuracy is 0.869 at L = 2, 0.909 at L = 3, 0.847 at L = 4 and 0.831 at L = 6, and L = 3 is optimal. Deeper circuits both train worse, because the federated averaging of many highly expressive local minima is destructive, and cost more to execute. The data are split across K = 9 clients. The reliable clients receive a balanced, class-covering slice, and the remaining data are Dirichlet(α = 0.5)-partitioned among the noisy clients, which models a deployment with a few well-calibrated reference sites and many noisy, heterogeneous edge clients. Device heterogeneity is modeled per client by a shot budget Mk and a depolarizing rate pk , and we study four heterogeneity conditions, listed in Table I, which independently vary the shot scarcity and the noise strength of the unreliable clients. This 2 × 2 design tests the prediction of Sec. III E that the precision (28) is only linear in Mk and quartic in (1 − pk ), and the noise rate rather than the shot budget should control the size of the QFedQGT advantage. Noisy clients additionally carry a coherent miscalibration, which makes Uk ̸= 0, and a readout-induced label corruption of a fraction 0.5 pk of their local labels, and their reported updates are therefore biased as well as statistically noisy. The complete hyperparameters, the exact noise profiles, the accuracy and error-bar definitions, and the fair-comparison protocol are collected in Appendix D.

10 TABLE I. The four device-heterogeneity conditions. The reliable clients always have Mk = 1000 shots and pk = 0, and the entries below are the shot budgets and depolarizing rates drawn for the unreliable clients. The last column is the closed-form weight ratio ϱnoisy /ϱgood of Eq. (30) for the most heavily discounted client of each condition, that is, the client with the highest depolarizing rate and the smallest shot budget. Panels (a)–(d) of Fig. 2 correspond to these rows. Panel Regime Noisy shots Mk Noisy rate pk ϱnoisy /ϱgood (a) few shots, strong noise {100, 200, 300} {0.4, 0.5, 0.6} 1.1 × 10−3 (b) more shots, strong noise {500, 600, 700} {0.4, 0.5, 0.6} 5.6 × 10−3 (c) few shots, weak noise {100, 200, 300} {0.1, 0.2, 0.3} 1.8 × 10−2 (d) more shots, weak noise {500, 600, 700} {0.1, 0.2, 0.3} 9.2 × 10−2

TABLE II. Accuracy on the reimei-E emulator (4000 shots, averaged over three seeds) for all four device-heterogeneity conditions of Table I. “Final” is the round-60 accuracy with its cross-seed standard error, “Mean” the average over the 23 evaluated rounds, “Std” the standard deviation across those rounds (a round-to-round stability measure, smaller is better), and “Worst” the lowest accuracy attained at any evaluated round. Best in each block in bold, with ties both bolded. Condition Method Final Mean Std Worst (a) few shots, QFedAvg 0.803 ± 0.009 0.774 0.040 0.703 strong noise QFedFisher 0.783 ± 0.007 0.738 0.039 0.667 QFedQGT 0.870 ± 0.023 0.814 0.044 0.753 (b) more shots, QFedAvg 0.810 ± 0.017 0.779 0.050 0.670 strong noise QFedFisher 0.807 ± 0.003 0.751 0.065 0.610 QFedQGT 0.843 ± 0.009 0.815 0.049 0.733 (c) few shots, QFedAvg 0.853 ± 0.012 0.803 0.054 0.700 weak noise QFedFisher 0.843 ± 0.026 0.767 0.082 0.597 QFedQGT 0.853 ± 0.017 0.818 0.045 0.737 (d) more shots, QFedAvg 0.863 ± 0.024 0.802 0.048 0.730 weak noise QFedFisher 0.860 ± 0.023 0.781 0.072 0.630 QFedQGT 0.870 ± 0.006 0.811 0.042 0.743 Round-averaged over (a)–(d) QFedAvg 0.790 QFedFisher 0.759 QFedQGT 0.815

We report the emulator evaluation. At each of 23 communication rounds, spaced densely from round 1 and reaching round 60, the current global model is dispatched to the trapped-ion emulator reimei-E of the reimei quantum computer [26] through Quantinuum Nexus. For each held-out test digit the three measurement settings Z, X, Y are executed with 4000 shots each, the Pauli expectations (23) are reconstructed from the returned counts, and the label is the arg max of the readout (24). The shot count was chosen from a measured shot–accuracy curve. Relative to the infinite-shot limit, the accuracy loss caused by finite readout statistics is 0.109 at 100 shots, 0.030 at 500 shots and 0.004 at 4000 shots, and at 4000 shots the reported numbers are essentially free of readout-statistics bias. The whole comparison is averaged over three independently seeded federations, and the error bars are the standard error of the accuracy across the three seeds. Because the submitted job specification is identical for the emulator and the physical device, the same experiment can be dispatched to reimei by changing only the backend name.

Results

Figure 2 and Table II report the emulator accuracy against the communication round for all four conditions. Three statements summarize the outcome, and each follows from the mixed-state framework of Sec. III. First, QFedQGT attains the highest round-averaged accuracy in every one of the four conditions, and its final-round accuracy is the highest in conditions (a), (b), and (d) and equal to that of QFedAvg in condition (c). Averaged over the four conditions its round-averaged accuracy is 0.815, against 0.790 for QFedAvg and 0.759 for QFedFisher, and its final-round accuracy is 0.859, against 0.832 and 0.823. Second, QFedQGT is the most robust of the three in the worst-round sense. Its worst evaluated round lies between 0.733 and 0.753 and exceeds the worst round of both baselines in every condition, and its round-to-round standard deviation is the smallest in three of the four conditions. Third, the size of the advantage is governed by the depolarizing rate and the shot budget. It is largest in condition (a), where the devices are the noisiest and the shots the scarcest, and smallest in condition (d), where the devices are the most reliable, as Eq. (28) predicts. We analyse these three points in turn. a. (i) Consistent advantage and few-round convergence. QFedQGT reaches its accuracy plateau within the first several communication rounds and retains the highest round-averaged accuracy throughout the evaluated window. In condition (a) the final-round accuracy of QFedQGT (0.870 ± 0.023) exceeds that of QFedAvg (0.803 ± 0.009) by

0.8

0.8 Test accuracy

(b) 1.0

Test accuracy

(a) 1.0

0.6

0.4

0.2

0.0

20 30 40 Communication round

0.4

0.2

QFedAvg QFedFisher QFedQGT

0.6

0.0

0.8

0.8 Test accuracy

(d) 1.0

Test accuracy

0.6

0.4

0.2

0.0

QFedAvg QFedFisher QFedQGT

20 30 40 Communication round

QFedAvg QFedFisher QFedQGT

20 30 40 Communication round

0.6

0.4

0.2

0.0

QFedAvg QFedFisher QFedQGT

20 30 40 Communication round

FIG. 2. Test accuracy against the communication round on MNIST-10 (n = 6 qubits, L = 3 layers, K = 9 clients), evaluated on the reimei-E trapped-ion emulator with 4000 shots per circuit and averaged over three seeds, with error bars the cross-seed standard error. Panels (a)–(d) are the four device-heterogeneity conditions of Table I, namely (a) few shots and strong noise, (b) more shots and strong noise, (c) few shots and weak noise, (d) more shots and weak noise. QFedQGT (solid, red) attains the highest round-averaged accuracy in every panel. The two baselines are closest to it in the weak-noise panels (c), (d) and fall further behind in the strong-noise panels (a), (b). The top and bottom rows differ only in the noise rate, and the left and right columns only in the shot budget, and the vertical separation is the larger of the two, as Eq. (28) predicts.

√ 0.067, which is 2.7 times the combined standard error 0.0232 + 0.0092 = 0.025, and in condition (b) the separation from the stronger baseline is 1.7 combined standard errors. Theorem 1 accounts for the early plateau through the geometric factor (1 − cηµ)T , in which the round count to reach suboptimality ϵ is Tϵ ≃ log(1/ϵ)/(cηµ), and two mechanisms of the mixed-state framework enlarge cη. First, b k + ε11)−1 in Eq. (13) rescales the update by the inverse curvature of the state manifold. the Bures preconditioner (G The Euclidean gradient is dominated by the stiff directions of GPure and is exponentially small along the flat ones, the barren-plateau pathology [31, 32], and preconditioning equalizes the two, which reduces the effective condition number λmax /λmin and, through η0 = min{1/L, cλ2min /(2Lβ 2 )}, permits a larger stable server step. Second, and specific to the federated setting, the precision weights (21) remove the unreliable clients from the aggregate at t = 0 rather than waiting for them to be averaged out. In the early rounds the noisy clients have biased updates, in that their labels are corrupted P at rate 0.5 pk and their measured gradients are attenuated by κ(pk ), and under QFedAvg the aggregated direction k pk dk has a systematic component pointing away from the descent direction, which reduces the alignment constant c of (A6). b. (ii) The noise rate and the shot budget control the gap. This comparison provides a direct test of the theory, and the 2 × 2 design of Table I was constructed for it. From Eqs. (26), (27) and (28), with N = 26 = 64 and γ = 3,

12 the weight ratio between an unreliable client and a reliable one is M κ(p)2 (1 − p)2 1 ϱnoisy · , = · 2 ϱgood κ(0) Mgood 1 + γp2

(30)

which is linear in the shot ratio M/Mgood and scales as κ(p)2 (1 − p)2 ≃ (1 − p)4 in the noise rate. Evaluated for the most heavily discounted client of each condition (last column of Table I), it gives 1.1 × 10−3 in (a), 5.6 × 10−3 in (b), 1.8 × 10−2 in (c) and 9.2 × 10−2 in (d). Raising the noise rate from {0.1, 0.2, 0.3} to {0.4, 0.5, 0.6} reduces the trust of the client by more than an order of magnitude, while a fivefold increase in its shot budget recovers about a factor of five. The measured accuracies follow this ordering. The round-averaged advantage of QFedQGT over QFedAvg is 0.040 in (a), 0.036 in (b), 0.015 in (c) and 0.009 in (d), decreasing monotonically as the devices become more reliable. Moving down a column of Fig. 2 (changing the noise) changes the advantage more than moving across a row (changing the shots) does, in agreement with the quartic-versus-linear dependence of Eq. (28). Because κ(p) is a Bures-metric quantity with no pure-state counterpart, this scaling is an experimental signature of the mixed-state geometry that a pure-state QGT or a classical Fisher matrix cannot produce. The adaptive floor (22) raises λfl toward λmax as the pool becomes reliable, and in the weak-noise conditions the rule approaches data-size averaging and the three methods converge, the parity that Proposition 1 requires in the homogeneous limit. c. (iii) Stability and the bound on the aggregated variance. Table II shows that QFedQGT has the smallest roundto-round standard deviation in conditions (b), (c), and (d), between 0.042 and 0.049, whereas QFedFisher reaches 0.082. This is the empirical signature of the error floor in Eq. (29). The floor contains P P the aggregated variance, which for a general weighting is Tr( k Ak Σk A⊤k ). For QFedAvg (Ak = pk 11) this equals k p2k Tr Σk and is dominated by 2 the noisiest client, because Tr Σk = vk Tr G−2 k grows as vk = 1/[Mk (1 − pk ) ], which for the worst client of condition (a) is 1/(100 × 0.16) = 6.3 × 10−2 against 10−3 for a reliable client, a factor of 62. Averaging cannot escape this, because the term p2k Tr Σk of a single bad client persists no matter how many good clients there Pare. −1 The precision weighting escapes it. By Eq. (18) the aggregated variance reduces to Tr[( ], and since k Λk ) P Λ ⪰ Λ for every k, monotonicity of the inverse on positive-definite matrices gives j k j Tr

h P

j Λj

−1 i

≤ Tr Λ−1 k

for every single client k.

(31)

The aggregated variance is therefore at most that of the best client alone and is never inflated by the bad ones, a statement with no counterpart for QFedAvg. Together with Proposition 1, this is the content of the observed stability, in that the round-to-round fluctuations of QFedQGT are smaller because its error floor is the smallest attainable by any unbiased linear aggregation of the client directions. d. (iv) Why QFedFisher is harmed by noise. QFedFisher is the weakest of the three methods in round-averaged accuracy (0.759 averaged over the four conditions), and its round-averaged accuracy is below that of plain averaging in every condition. This is a structural effect, and the mixed-state framework diagnoses it. QFedFisher weights clients bk ∥, whose expectation is by the norm of the diagonal empirical Fisher statistic, wk ∝ pk ∥b gk ⊙ g 2 2 E gbk,i = ∂i fk + vk ,

vk =

1 , Mk (1 − pk )2

(32)

in which the shot-noise variance is added to the statistic. Because vk increases as the shot budget falls and as the depolarizing rate rises, the QFedFisher weight is a monotonically increasing function of the client unreliability, whereas the achievable precision prescribed by the quantum Cramér–Rao bound, Tr Λk = κ(pk )2 ∥GPure ∥2F /vk , is a monotonically decreasing function of the same quantity. The two prescriptions are reciprocals of one another, and QFedFisher up-weights the clients that QFedQGT down-weights. The weight report of Appendix D confirms this in condition (a), where QFedFisher assigns the noisy clients a mean weight 1.46 times that of the reliable clients, while QFedQGT assigns them 0.042 times. A further weakness is that the QFedFisher weights are re-estimated every round from a fresh minibatch under shot noise, and they fluctuate from round to round, with a mean per-client standard deviation of 0.044 in condition (a), whereas the QFedQGT and QFedAvg weights are deterministic functions of (Mk , pk , κ, rk ). This fluctuation is consistent with QFedFisher having the largest round-to-round accuracy standard deviation in three of the four conditions. QFedAvg avoids the weight inversion, because its weights are noise-blind, and, weighting purely by data size, P 2it cannot distinguish a reliable client from an unreliable one, and it is held at the QFedAvg variance floor k pk Tr Σk of point (iii). e. (v) The advantage is a mixed-state effect. All of the mechanisms above vanish in the pure-state limit. If the clients were noiseless one would have κ(pk ) = 1, Uk = 0 and vk = 1/Mk , hence ϱk ∝ Mk , and with equal shot budgets

13 wk → pk and QFedQGT degenerates to QFedAvg, while the Bures preconditioner degenerates to the pure-state QNG of Eq. (7). Every number reported above originates in the mixed-state extension. The attenuation κ(pk ) of Eq. (26) is a Bures-metric quantity that has no pure-state counterpart, and the incompatibility temper 1/(1 + γrk ) with rk = ∥Uk ∥2F /∥Gk ∥2F is a mean-Uhlmann-curvature quantity that is zero for pure states and for any commuting, hence classical-Fisher, family. Consistently, the weak-noise conditions (c) and (d), where κ(p) → 1 and rk → 0, are the ones in which the three methods come closest together, the parity that Proposition 1 requires in the homogeneous limit, while the strong-noise conditions, where the mixed-state corrections are largest, are where the separation is widest. The experiment therefore shows that the piece of geometry invisible to both the pure-state QGT and the classical Fisher matrix is the piece that carries the gain.

Contribution of the individual components

Three controlled comparisons isolate the contribution of each component. First, replacing the exact block-diagonal Bures metric by its diagonal truncation lowers accuracy and removes the Uhlmann incompatibility signal, whereas the block-diagonal estimator stays close to the full metric at O(L) cost, in agreement with the bound of Appendix B. Second, at low noise the pure-state QGT (that is, QNG) and the Bures metric are equivalent, while at moderate and heterogeneous noise the Bures preconditioner is more accurate, which locates the gain in the mixed-state correction, as argued in point (v) above. Third, including the incompatibility term C(ωkUhl ) in (21) improves robustness under local and coherent noise, where Uk ̸= 0, which isolates the role of the imaginary part of the mixed-state QGT.

VI.

CONCLUSION AND OUTLOOK

We have reformulated geometric QFL for the noisy regime in which it runs. Recognizing that the real part of the pure-state QGT is the QFIM, and hence that pure-state QGT preconditioning reproduces the QNG, we extended the geometry to mixed states, where the Bures metric and the mean Uhlmann curvature are distinct from both the QFIM and the classical Fisher matrix. The Bures part serves as a reparametrization-invariant local preconditioner, and the Uhlmann part enters, through the quantum Cramér–Rao incompatibility correction, an achievable-precision aggregation rule that minimizes the residual variance (Proposition 1) and is robust to device heterogeneity. We replaced the diagonal approximation by an exact block-diagonal estimator with bounded truncation error, and we derived the closed-form depolarizing attenuation κ(p). On the ten-class MNIST task evaluated on the reimei-E trapped-ion emulator under four device-heterogeneity conditions and averaged over three seeds, QFedQGT attained the highest round-averaged accuracy in every condition, with a round-averaged accuracy of 0.81 and a final-round accuracy of 0.86, against 0.79 and 0.83 for federated averaging, and with the highest worst-round accuracy throughout. The size of the advantage tracked the closed-form weight ratio κ(p)2 (1 − p)2 M/[Mgood (1 + γp2 )], largest where the theory predicts the strongest down-weighting and narrowing toward parity where the devices are reliable. Because the submitted circuits and workflow are identical to those required by the physical device, the same evaluation can be dispatched to the reimei hardware without code changes. Several directions extend this work. The block-diagonal estimator can be generalized to include the inter-layer components of the mixed-state geometry within a controlled measurement budget, which would tighten the truncation bound of Appendix B and quantify the residual bias of the layerwise approximation on deeper ansätze. The achievableprecision aggregation admits combination with quantum error correction, in which the precision matrix Λk reflects the logical rather than the physical error rate, and the attenuation κ(pk ) is replaced by the residual logical error after decoding. A demonstration on the reimei device at a larger qubit count and class number would test the framework beyond the emulation reported here and probe the regime in which coherent and correlated errors, for which Uk ̸= 0, dominate the depolarizing contribution.

ACKNOWLEDGMENTS

This work was supported by JST Moonshot R&D Grant No. JPMJMS2061, JST CREST Grant No. JPMJCR23I4, MEXT Q-LEAP Grant No. JPMXS0120319794, JST ASPIRE Grant Number JPMJAP2318, JST SPRING Grant Number JPMJSP2119, RIKEN Junior Research Associate Program, and RIKEN TRIP initiative (RIKEN Quantum).

14 Appendix A: Derivation of κ(p) for global depolarizing

We derive Eq. (26) from the definition (10) of the SLD quantum Fisher information, keeping every step explicit, and the result does not rely on the formula it establishes. Fix a client and a single noise level p ∈ [0, 1) and write the global-depolarizing state of the pure state |ψ⟩ = |ψ(θ)⟩ as ρ = (1 − p)|ψ⟩⟨ψ| +

p 11, N

N = 2n .

(A1)

Choose an orthonormal eigenbasis of ρ consistingPof |ψ⟩ together with any N − 1 vectors {|m⊥ ⟩} spanning the orthogonal complement |ψ⟩⊥ . Because 11 = |ψ⟩⟨ψ|+ m⊥ |m⊥ ⟩⟨m⊥ |, the state is diagonal in this basis with eigenvalues λψ = a ≡ (1 − p) +

p , N

λm⊥ = b ≡

p (N − 1-fold). N

(A2)

Both eigenvalues are strictly positive for p ∈ (0, 1), and every pair (m, n) has λm + λn > 0 and contributes in (10). Differentiating ρ with respect to θi , and using that p and N are constants, gives ∂i ρ = (1 − p) |∂i ψ⟩⟨ψ| + |ψ⟩⟨∂i ψ| . (A3) We adopt the gauge Re⟨ψ|∂i ψ⟩ = 0, which can always be imposed by a local phase choice and leaves ρ unchanged, hence ⟨ψ|∂i ρ|ψ⟩ = 2(1 − p) Re⟨ψ|∂i ψ⟩ = 0. The operator (A3) has no diagonal block and connects only |ψ⟩ with the complement. The only nonzero matrix elements of ∂i ρ are ⟨m⊥ |∂i ρ|ψ⟩ = (1 − p)⟨m⊥ |∂i ψ⟩,

⟨ψ|∂i ρ|m⊥ ⟩ = (1 − p)⟨∂i ψ|m⊥ ⟩,

(A4)

and the eigenvalue sum attached to each such pair is a + b = 1 − p + 2p/N . Insert these elements into (10). The pairs (m, n) = (m⊥ , ψ) and (ψ, m⊥ ) give identical real contributions, and Bur FijQ,SLD = 4gij =

4 X Re (1 − p)⟨∂i ψ|m⊥ ⟩ (1 − p)⟨m⊥ |∂j ψ⟩ , a+b m

(A5)

⊥

in which the factor 4P collects the 12 of (10) doubled by the two orderings of the pair. Using the completeness relation on the complement, m⊥ |m⊥ ⟩⟨m⊥ | = 11 − |ψ⟩⟨ψ|, the sum is X ⟨∂i ψ|m⊥ ⟩⟨m⊥ |∂j ψ⟩ = ⟨∂i ψ| 11 − |ψ⟩⟨ψ| ∂j ψ⟩ = QPure , (A6) ij m⊥

whose real part is Re QPure = gij . Therefore ij FijQ,SLD =

(1 − p)2 4(1 − p)2 Re QPure = F Q,Pure , ij 1 − p + 2p/N 1 − p + 2p/N ij

(A7)

where the last step uses the pure-state identity FijQ,Pure = 4 Re QPure (Sec. II B). This is Eq. (26) with κ(p) = ij (1 − p)2 /(1 − p + 2p/N ). The limits follow by inspection. One has κ(0) = 1, and for N ≫ 1 the term 2p/N is negligible, hence κ(p) ≃ (1 − p)2 /(1 − p) = 1 − p and κ(p)2 (1 − p)2 ≃ (1 − p)4 , the quartic suppression used in Sec. V. For the experiments, N = 64 and the values entering Eq. (30) are κ(0.1) = 0.897, κ(0.2) = 0.794, κ(0.3) = 0.691, κ(0.4) = 0.588, κ(0.5) = 0.485 and κ(0.6) = 0.382. □ Appendix B: Block-diagonal truncation error

We bound the error incurred by replacing the regularized full Bures metric by its block-diagonal (layerwise) approximation, tracking every constant. Write the regularized metric as X = G + ε11 and its block-diagonal counterpart as Y = B + ε11, where B = Bdiag(G(1) , . . . , G(L) ) collects the exact intra-layer blocks and E ≡ G − B collects the neglected inter-layer blocks. Both G and B are real symmetric and positive semidefinite, because they are Gram matrices of the layer generators, hence X and Y are symmetric with all eigenvalues ≥ ε > 0. Consequently X and Y are invertible and, in operator norm, X −1 ≤ ε−1 ,

Y −1 ≤ ε−1 .

(B1)

15 The second-resolvent identity, which for any two invertible matrices reads X −1 − Y −1 = X −1 (Y − X)Y −1 , gives, with Y − X = B − G = −E, X −1 − Y −1 = − X −1 E Y −1 .

(B2)

Taking the operator norm, and using submultiplicativity and (B1), X −1 − Y −1 ≤ X −1 ∥E∥ Y −1 ≤ ε−2 ∥E∥ .

(B3)

The natural-gradient direction computed with the two metrics therefore differs by X −1 − Y −1 ∇f ≤ X −1 − Y −1 ∥∇f ∥ ≤ ε−2 ∥E∥ ∥∇f ∥ ,

(B4)

which is controlled by the size ∥E∥ of the inter-layer coupling and is small for ansätze with geometrically local entanglement, where ∥E∥ decays with the separation between layers. Unlike the diagonal approximation, the blockdiagonal estimator retains every intra-layer matrix element, and hence the full antisymmetric incompatibility Uk of Eq. (11), exactly. □ Appendix C: Proof of Theorem 1

We give a self-contained proof. Throughout, Et [ · ] denotes conditional expectation given the iterate θt , E[ · ] the total expectation, and ∆t ≡ E[F (θt ) − F ∗ ]. The one-step update is X X ∇f (θ ) + ξ , Ak = 11, (C1) θt+1 = θt − η Ak dk,t , dk,t = G−1 k t k,t k k

with η = ηSrv , and by (A3) the noises satisfy Et [ξk,t ] = 0, Et [ξk,t ξ⊤ k,t ] = Vk , and independence across k. We write ∇F ≡ ∇F (θt ). Without loss of generality we take c ≤ 1 in (A6), because (A6) remains valid when c is decreased. a. Step 1 (descent inequality from smoothness). By the L-smoothness upper bound in (A1) applied to u = θt and v = θt+1 , and taking conditional expectation Et , hX i Lη 2 h X 2i Et [F (θt+1 )] ≤ F (θt ) − η ∇F⊤ Et Ak dk,t + Ak dk,t . Et 2 k

(C2)

Because Et [dk,t ] = G−1 k ∇fk (θt ), the alignment assumption (A6) gives hX i X 2 ∇F⊤ Et Ak dk,t = ∇F⊤ Ak G−1 ≥ c ∥∇F ∥ . k ∇fk

Step 2 (the first-order term).

(C3)

k 2

c. Step 3 (the second-order term). Decompose the squared norm into its mean and fluctuation through Et [∥Z∥ ] = P P 2 2 ∥Et [Z]∥ + Et ∥Z − Et [Z]∥ with Z = k Ak dk,t . The fluctuation is Z − Et [Z] = k Ak G−1 k ξk,t , and because the ξk,t are independent and zero-mean, the cross terms vanish, and h X X X 2i 2 −1 ⊤ + Tr Ak G−1 (C4) Et Ak dk,t = Ak G−1 k Vk Gk Ak . k ∇fk k

Bound on (I). Write ∇fk = ∇F + (∇fk − ∇F ) and use P 2 2 2 −1 k Ak Gk (∇fk − ∇F ). By ∥a + b∥ ≤ 2 ∥a∥ + 2 ∥b∥ , (I) ≤ 2

}

(I)

Ak G−1 k ∇F

k Ak = 11 to split

}

(II)

−1 k Ak Gk ∇fk =

Ak G−1 k (∇fk − ∇F )

−1 k Ak Gk

∇F +

(C5)

P P −1 −1 For the first piece, G−1 ≤ λ−1 min by (A5) and k ∥Ak ∥ ≤ β give k Ak Gk ∇F ≤ βλmin ∥∇F ∥, and this piece is k P 2 −1 2 −2 at most 2β λmin ∥∇F ∥ . For the second piece set wk = ∥Ak ∥ (hence kP wk ≤ β) and uP ∥∇fk − ∇F ∥. The k = λminP 2 triangle inequality followed by the weighted Cauchy–Schwarz inequality ( k wk uk ) ≤ ( k wk )( k wk u2k ) gives X 2 X X X 2 2 ≤ wk uk ≤ β wk u2k ≤ β 2 λ−2 ∥∇fk − ∇F ∥ . (C6) Ak G−1 min k (∇fk − ∇F ) k

16 By (A4),

k ∥∇fk − ∇F ∥

2 ≤ Kζ 2 , and the second piece is at most β 2 λ−2 min Kζ and, with its prefactor 2,

2β 2 Kζ 2 2β 2 2 2 2 ∥∇F ∥ + ζeff . (C7) , ζeff ≡ 2 λmin λ2min P P −1 Bound on (II). The matrices Ak are free subject to k Ak = 11, and (II)= Tr( k Ak Σk A⊤k ) with Σk = G−1 k Vk Gk . Minimizing P this quadratic formPunder the linear constraint is a matrix least-squares problem. Form the Lagrangian G = Tr( k Ak Σk A⊤k ) − Tr[Θ⊤ ( k Ak − 11)] with a multiplier matrix Θ. Stationarity ∂G/∂Ak = 2Ak Σk − Θ = 0 gives P P P 1 1 1 −1 , whence Ak = 21 ΘΣ−1 k Ak = 2 Θ k Λk = 11 fixes 2 Θ = ( k Λk ) k = 2 ΘΛk , and the constraint (I) ≤

Aopt = k

Λj

−1

Λk ,

(C8)

which is (18). The stationary point is the minimum, because the objective is convex in {Ak } (Σk ≥ 0). Substituting Aopt and using Λk Σk Λk = Λk (since Σk = Λ−1 k k ), h P h P −1 i −1 i P −1 P = Tr . (C9) (II)min = Tr j Λj k Λk k Λk j Λj d.

Step 4 (the one-step recursion).

Insert (C3), (C7) and (C9) into (C2),

P Lη 2 2 Lη 2 β 2 2 ∥∇F ∥ + ζeff + Tr ( k Λk )−1 . Et [F (θt+1 )] ≤ F (θt ) − cη − 2 λmin 2

(C10)

The hypothesis η ≤ η0 ≤ cλ2min /(2Lβ 2 ) implies Lη 2 β 2 /λ2min ≤ 12 cη, hence cη − Lη 2 β 2 /λ2min ≥ 21 cη. Applying the 2 Polyak–Lojasiewicz inequality (A2), ∥∇F ∥ ≥ 2µ (F (θt ) − F ∗ ), to the negative term, subtracting F ∗ from both sides, and taking total expectation, ∆t+1 ≤ (1 − cηµ) ∆t +

P Lη 2 2 ζeff + Tr ( k Λk )−1 . 2

(C11)

The contraction factor satisfies 0 ≤ 1−cηµ < 1. The strict upper because c, η, µ > 0. For the lower bound, bound holds 2 2 1 ∥∇F ∥ , hence ∥∇F ∥ ≤ 2L (F (θ) − F ∗ ), minimality of F ∗ together with (A1) gives F ∗ ≤ F θ − L1 ∇F ≤ F (θ) − 2L which combined with (A2) yields µ ≤ L, and with c ≤ 1 and η ≤ 1/L we obtain cηµ ≤ µ/L ≤ 1. e. Step 5 (unrolling the recursion). Inequality (C11) has the form ∆t+1 ≤ q ∆t + b with q = 1 − cηµ ∈ [0, 1) and P PT −1 2 2 constant b = Lη2 ζeff + Tr[( k Λk )−1 ] ≥ 0. Iterating from t = 0 to T − 1 and bounding the geometric sum t=0 q t by its infinite limit 1/(1 − q) = 1/(cηµ), ∆ T ≤ q T ∆0 +

P Lη 2 b = q T ∆0 + ζeff + Tr ( k Λk )−1 , 1−q 2cµ

(C12)

which is Eq. (29) with ∆0 = F (θ0 ) − F ∗ . Every constant (L, µ, c, β, λmin , ζ, Λk ) is the one introduced in Assumption 1 and Eqs. (15)–(21), and no step used the conclusion, and the argument is non-circular. □ Appendix D: Experimental configuration, evaluation, and emulation

Model and data. The shared dressed classifier has n = 6 qubits and L = 3 layers, giving dq = 3nL = 54 quantum parameters, with each layer re-uploading the data through Ry (xj ), applying a trainable Rz Ry Rz block per qubit, and a linear CNOT entangler (Fig. 1). The readout measures the Mf = 3n + n(n − 1)/2 = 33 Pauli observables of Eq. (23) in three settings and feeds them to the shared tanh head (24) with H = 256 hidden units. The task is the ten-class set {0, . . . , 9} of the full MNIST database [46], loaded through tf.keras.datasets.mnist, with pixels normalized to [0, 1], and 3000 training images per class (30,000) are drawn from the training split and 200 test images per class (2000) from the test split. Principal-component analysis reduces the 784 pixels to nf = nL = 18 components, which are standardized and scaled by 0.4 to keep the re-uploading rotations in their near-linear regime, with the PCA basis and the standardizer fitted on the training split only. Federated and noise configuration. There are K = 9 clients with full participation each round. A fraction 0.4 of them (four clients) are reliable and receive a balanced class-covering slice, and the remaining five are unreliable and receive the rest of the data under a Dirichlet(α = 0.5) partition, in which a smaller α is more non-IID. Each

17 TABLE III. Mean aggregation weight given to the noisy and to the reliable clients in condition (a), and the mean per-client round-to-round standard deviation of the weights. A ratio below one indicates down-weighting of the noisy clients. Method noisy reliable ratio weight jitter QFedAvg 0.120 0.100 1.20 0.000 QFedFisher 0.129 0.088 1.46 0.044 QFedQGT 0.010 0.238 0.042 0.000

client performs τ = 20 local minibatch steps per round at batch size 32, capped at E = 50 epochs, with learning (t) rate ηLoc = 0.02 decayed as ηLoc = ηLoc /(1 + 0.02 t), and softmax inverse temperature βout = 3. The server step is ηSrv = 0.7 and the number of rounds is T = 60. The QFedQGT hyperparameters are the Tikhonov shift ε = 0.05, the incompatibility temper γ = 3, and the adaptive floor of Eq. (22) with λ0 = 0.05, λmax = 0.20, plo = 0.10 and phi = 0.30. Reliable clients have Mk = 1000 shots and pk = 0, and unreliable clients draw Mk and pk from the sets in Table I. The measured expectation values of a noisy client are attenuated as ⟨P ⟩ → (1−pk )⟨P ⟩ and estimated from Mk binomial samples, and the effective gradient variance is vk = 1/[Mk (1 − pk )2 ] as in Eq. (27). Noisy clients additionally carry a coherent gate miscalibration of magnitude 0.3 pk along a client-specific direction, which makes Uk ̸= 0, and we use the leading-order proxy rk = p2k , and a readout-induced label corruption in which a fraction 0.5 pk of their local training labels are randomized. Local gradients are computed by exact reverse-mode (adjoint) differentiation of the statevector, numerically identical to the parameter-shift rule (verified to 2 × 10−15 ) at cost O(#gates) rather than O(2dq ) circuit evaluations [47], and the device-noise model above is applied on top, and the optimizer sees the noisy, finite-shot gradients that hardware would return. Aggregation weights across clients. Table III reports the mean aggregation weight that each method assigns the noisy and the reliable clients in condition (a), together with the round-to-round standard deviation of the weights. QFedQGT down-weights the noisy clients by a factor 0.042 relative to the reliable ones, QFedFisher up-weights them by a factor 1.46, and QFedAvg is close to uniform. The QFedAvg and QFedQGT weights are deterministic, while the QFedFisher weights fluctuate from round to round, with a mean per-client standard deviation of 0.044. Held-fixed quantities for a fair comparison. The three methods are run on identical ansatz, readout head, data partition, per-client noise profiles, random seeds, initial parameters, and evaluation set, and they differ only in the local update rule and the aggregation weights. QFedAvg uses data-size weights with a plain Adam update, QFedFisher uses layerwise diagonal classical-Fisher weights, and QFedQGT uses Eqs. (13) and (21). All client-side randomness is seeded deterministically from the pair (round, client), and each run is reproducible and any accuracy difference is attributable to the aggregation and geometry rather than to a different data split or initialization. Definition of accuracy and of the error bars. For each test digit the trained global model is executed on the emulator in the three measurement settings, the Pauli expectations (23) are reconstructed from the returned counts, each ⟨P ⟩ being the parity-weighted mean of the bitstrings, the class scores are formed by the readout (24), and the predicted label is their arg max. The reported accuracy is the fraction of the test digits whose predicted label matches the ground truth, averaged over three independently seeded federations. The error bars are the standard error of the accuracy across the three seeds, which measures the run-to-run variability that arises from the data partition, the device assignment, and the initialization, and which is the variability an experimentalist would report over repeated deployments. The emulator, and the path to hardware. The reimei-E emulator is the noise-inclusive software model of the reimei trapped-ion processor [26], reproducing the device native gate set, its all-to-all connectivity, and its calibrated error model, and accepting the same job specification as the physical machine. Circuits are constructed and compiled with pytket [48] and submitted asynchronously through Quantinuum Nexus, and the submission is resumable, and a network interruption never loses completed work. Because the submitted workflow is identical to a hardware submission, with only the backend name changing from reimei-E to reimei, the emulator results are a preview of on-device behaviour and the same experiment can be dispatched to the physical processor without any change to the code.

[1] B. McMahan, E. Moore, D. Ramage, S. Hampson, and B. A. y. Arcas, in Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, Proceedings of Machine Learning Research, Vol. 54, edited by A. Singh and J. Zhu (PMLR, 2017) pp. 1273–1282. [2] M. Benedetti, E. Lloyd, S. Sack, and M. Fiorentini, Quantum Sci. Technol. 4, 043001 (2019). [3] M. Cerezo, A. Arrasmith, R. Babbush, S. C. Benjamin, S. Endo, K. Fujii, J. R. McClean, K. Mitarai, X. Yuan, L. Cincio, and P. J. Coles, Nat. Rev. Phys. 3, 625 (2021).

18 [4] M. Schuld, R. Sweke, and J. J. Meyer, Phys. Rev. A 103, 032430 (2021). [5] S. Y.-C. Chen and S. Yoo, Entropy 23, 460 (2021). [6] D. C. Nguyen, M. R. Uddin, S. Shaon, R. Rahman, O. Dobre, and D. Niyato, arXiv:2508.15998 10.48550/arXiv.2508.15998 (2025). [7] S. Sai, A. Sawaika, P. Singh, and R. Buyya, arXiv:2510.17642 10.48550/arXiv.2510.17642 (2025). [8] A. S. Bhatia and S. Kais, arXiv:2507.17580 10.48550/arXiv.2507.17580 (2025). [9] S. I. Nanayakkara and S. R. Pokhrel, arXiv:2512.03363 10.48550/arXiv.2512.03363 (2025). [10] J. Han, N. S. DiBrita, D. Leeds, J. Li, J. Ludmir, and T. Patel, arXiv:2510.06228 10.48550/arXiv.2510.06228 (2025). [11] A. Pokharel, R. Rahman, S. Shaon, T. Morris, and D. C. Nguyen, arXiv:2508.20310 10.48550/arXiv.2508.20310 (2025). [12] J. P. Provost and G. Vallee, Commun. Math. Phys. 76, 289 (1980). [13] R. Cheng, arXiv:1012.1337 10.48550/arXiv.1012.1337 (2010). [14] A. Gao, N. Nagaosa, N. Ni, and S.-Y. Xu, arXiv:2508.00469 10.48550/arXiv.2508.00469 (2025). [15] T. Liu, X.-B. Qiang, H.-Z. Lu, and X. C. Xie, Natl. Sci. Rev. 12, nwae334 (2025). [16] J. Stokes, J. Izaac, N. Killoran, and G. Carleo, Quantum 4, 269 (2020). [17] J. Liu, H. Yuan, X.-M. Lu, and X. Wang, J. Phys. A: Math. Theor. 53, 023001 (2019). [18] D. Bures, Trans. Amer. Math. Soc. 135, 199 (1969). [19] A. Uhlmann, Lett. Math. Phys. 21, 229 (1991). [20] D. Petz, Linear Algebra Appl. 244, 81 (1996). [21] A. Carollo, D. Valenti, and B. Spagnolo, Phys. Rep. 838, 1 (2020). [22] C. W. Helstrom, J. Stat. Phys. 1, 231 (1969). [23] C. W. Helstrom, Quantum Detection and Estimation Theory, Vol. 123 (Academic Press, New York, 1976). [24] A. S. Holevo, Probabilistic and Statistical Aspects of Quantum Theory (North-Holland, Amsterdam, 1982). [25] M. G. A. Paris, Int. J. Quant. Inf. 7, 125 (2009). [26] Quantinuum reimei-E, accessed August 8–9, 2026. [27] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, Phys. Rev. A 98, 032309 (2018). [28] M. Schuld, V. Bergholm, C. Gogolin, J. Izaac, and N. Killoran, Phys. Rev. A 99, 032331 (2019). [29] M. V. Berry, Proc. R. Soc. A 392, 45 (1984). [30] S.-i. Amari, Neural Comput. 10, 251 (1998). [31] J. R. McClean, S. Boixo, V. N. Smelyanskiy, R. Babbush, and H. Neven, Nat. Commun. 9, 4812 (2018). [32] S. Wang, E. Fontana, M. Cerezo, K. Sharma, A. Sone, L. Cincio, and P. J. Coles, Nat. Commun. 12, 6961 (2021). [33] S. Imai, J. Yang, and L. Pezzè, Phys. Rev. Lett. 136, 150801 (2026). [34] S. Imai, J. Yang, and L. Pezzè, arXiv:2602.12097 10.48550/arXiv.2602.12097 (2026). [35] D. G. Brennan, Proceedings of the IRE 47, 1075 (1959). [36] T. Li, A. K. Sahu, M. Zaheer, M. Sanjabi, A. Talwalkar, and V. Smith, in Proceedings of Machine Learning and Systems, Vol. 2, edited by I. Dhillon, D. Papailiopoulos, and V. Sze (2020) pp. 429–450. [37] A. Mari, T. R. Bromley, J. Izaac, M. Schuld, and N. Killoran, Quantum 4, 340 (2020). [38] A. Pérez-Salinas, A. Cervera-Lierta, E. Gil-Fuster, and J. I. Latorre, Quantum 4, 226 (2020). [39] A. Mari, T. R. Bromley, and N. Killoran, Phys. Rev. A 103, 012405 (2021). [40] J. J. Meyer, Quantum 5, 539 (2021). [41] J. L. Beckey, M. Cerezo, A. Sone, and P. J. Coles, Phys. Rev. Res. 4, 013083 (2022). [42] K. M. Nakanishi, K. Fujii, and S. Todo, Phys. Rev. Res. 2, 043158 (2020). [43] L. Bottou, F. E. Curtis, and J. Nocedal, SIAM Review 60, 223 (2018). [44] H. Karimi, J. Nutini, and M. Schmidt, in Machine Learning and Knowledge Discovery in Databases, edited by P. Frasconi, N. Landwehr, G. Manco, and J. Vreeken (Springer International Publishing, Cham, 2016) pp. 795–811. [45] J. Wang, Q. Liu, H. Liang, G. Joshi, and H. V. Poor, in Advances in Neural Information Processing Systems, Vol. 33, edited by H. Larochelle, M. Ranzato, R. Hadsell, M. Balcan, and H. Lin (Curran Associates, Inc., 2020) pp. 7611–7623. [46] Y. LeCun, L. Bottou, Y. Bengio, and P. Haffner, Proceedings of the IEEE 86, 2278 (1998). [47] T. Jones and J. Gacon, arXiv:2009.02823 10.48550/arXiv.2009.02823 (2020). [48] S. Sivarajah, S. Dilkes, A. Cowtan, W. Simmons, A. Edgington, and R. Duncan, Quantum Sci. Technol. 6, 014003 (2020).

Quantum Federated Learning Based on Bures--Uhlmann Geometry for Heterogeneous Noisy Clients

Analyze & explore related research in Conceptio

Related documents