Structural Conditions for Distributed Quantum Advantage

Structural Conditions for Distributed Quantum Advantage Sabina Drăgoi,1, 2, ∗ Marı́a Gragera Garcés,3, † and Lirandë Pira4, ‡

arXiv:2610.04393v1 [quant-ph] 3 Oct 2026

1

Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland 2 IBM Research, Säumerstrasse 4, CH-8803 Rüschlikon, Switzerland 3 Quantum Software Lab, University of Edinburgh, Edinburgh, United Kingdom 4 Center for Quantum Technologies, National University of Singapore, Singapore Circuit cutting runs a quantum computation on processors smaller than the circuit, at the price of classical knitting whose cost grows exponentially with the number of cut gates. Keeping this cost affordable is necessary but not sufficient for an advantage, since knitting classically easy subcircuits is itself classically easy. We formulate three requirements for distributed quantum advantage under a stated budget: affordable knitting, classical difficulty that survives the cut, and, for variational circuits, resolvable gradients. We prove that under stated assumptions knitting stays affordable when subcircuits grow behind a bounded interface and becomes exponentially expensive when fixed-width subcircuits multiply. We turn the requirements into screening criteria and apply them to eighteen circuit families. We identify finite local-depth circuits with bounded interfaces as a setting in which the requirements can coexist, with hardness established only in the worst case. As a classically verifiable proof of principle, we knit one gate joining two field-perturbed toric-code patches on an IBM Nighthawk processor for up to 142 spins, and confirm that the reconstruction can at least resolve the contact correlation beyond independent execution for up to 98 qubits.

I.

INTRODUCTION

Many applications of quantum computing require more qubits than a single processor currently provides [1–4]. As in classical distributed computing [5, 6], a natural way to scale up is to distribute the computation across several quantum chips [7–9]. Quantum links between processors have been demonstrated [10], while circuit cutting [11– 16] offers an alternative requiring no quantum link. It removes the gates that act across a chosen partition of the qubits, runs the resulting subcircuits separately, and classically knits their measurement outcomes together. Circuit cutting thus trades the width of a processor for classical post-processing work. However, this approach faces two obstacles. Knitting requires a number of samples that grows exponentially with the number of removed gates [17–20]. Each removed gate is replaced by a signed combination of local operations, and averaging out the resulting cancellations multiplies the number of samples by a constant factor for every gate. A cheap cut is not enough either. If the subcircuits are easy to simulate classically, knitting their results together is easy as well [11, 19], and the quantum hardware offers no advantage [21]. A circuit that promises a quantum advantage is therefore not automatically a good candidate for distributed execution. Variational algorithms add a third requirement. They exploit the expressibility of parametrized circuits, which can prepare useful states with few gates, provided their parameters are well optimized [22–25]. These parameters are usually trained using gradient-based methods that scale only if the gradients remain resolvable as the system

∗ sdragoi@phys.ethz.ch † m.gragera.garces@ed.ac.uk ‡ lpira@nus.edu.sg

grows, and at random initialization they often do not. In a barren plateau, gradient variances decay exponentially with the number of qubits, so resolving a gradient takes exponentially many measurements [26–30]. Keeping circuits shallow can bring their observables within reach of classical simulation in the regimes studied in Refs. [31–33]. The simulation guarantees depend on the architecture, ensemble, accuracy, and access to input-state data; successful simulations of large experimental circuits provide further examples [34, 35]. For architectures designed to balance trainability and expressivity, the encoding, readout, and required accuracy must also be specified when assessing classical complexity [36]. Our earlier work made this conflict precise for a one-dimensional brickwork circuit split into two blocks joined by a fixed number of gates [37]. In that setting, for area-law input states and local costs at random initialization, avoiding a barren plateau requires depth O(log n), whereas escaping the matrix-product-state simulation criterion used there requires a superpolynomial bond dimension, which in turn requires depth ω(log n). This paper asks which circuit families are worth distributing. First, we identify three requirements. Knitting must remain affordable as the system grows, which we call Cut. The tasks left to each subcircuit must remain classically hard, which we call Sim, since otherwise the knitted result is classically easy too. For variational circuits, gradients must remain resolvable, which we call Train. We turn each requirement into criteria that can be checked for a given circuit family and resource budget. Assessed together, the requirements locate where a distributed advantage can reside. An efficient cut whose subcircuit tasks are classically easy is classically easy as a whole (Theorem 1). For the cutting constructions and target states we consider, growing the subcircuits behind a bounded or logarithmically growing interface keeps knitting affordable, whereas growing the number of

2 Requirements Sec. II

Screening criteria Table I

Cut affordable knitting, Γcut ≤ poly(n)

C1 interface κ and norm Γcut C2 seam entropy S1/2

Sim hardness survives the cut

S1 hard branch tasks S2 light cones S3 task type

Train resolvable gradients

T1 local depth T2 guarantee g T3 gradient variance

General results Prop. 1 seam entropy sets a floor on Γcut Thm. 1 knitting easy branch tasks is easy Thm. 2 growing subcircuits, bounded interface

eighteen circuit families Tables II and III

Single-bridge application Sec. IV

Cheaply cuttable, still hard row 11c, and 12b if S1 holds

Two toric-code patches C1 ✓ cost flat in N C2 ✓ S1/2 ≤ 1 bit T1 ✓ bounded local depth T3 ✓ no exponential decay at tested sizes S1 ? hardness open

Cuttable, no advantage rows 7, 8a, and 11b Advantage, not cuttable row 11a Optimally cut, unaffordable rows 8b and 12a

Hardware, IBM Nighthawk contact correlation up to N = 98 knitted energies up to N = 142

Structural failure rows 1–6

FIG. 1. How three requirements for distributed quantum advantage sort eighteen circuit families. The three requirements for distributed quantum advantage (left) become eight screening criteria, and three general results underpin the knitting and hardness criteria (bottom). Applied to eighteen circuit families, the criteria sort them into classes. Every class except the target is colored by the requirement it fails, with the structural failures shaded from Train to Cut, the two they fail, where a structural failure means a missing guarantee rather than a proven absence of advantage. Only the highlighted class combines a cheap cut with hardness inside a subcircuit, and it is the class that distributed finite local-depth circuits (FLDCs) aim for. The single-bridge application meets the knitting and trainability criteria at the tested sizes, where its branch tasks are still classically tractable, so the hardness criterion S1 remains open beyond them. The dashed box summarizes the device runs.

fixed-width subcircuits makes it exponentially expensive (Theorem 2). An advantage must therefore come from subcircuits that grow behind a bounded interface and remain classically hard. Second, we apply the criteria to eighteen circuit families and regimes from the literature (Table II), from standard variational ground-state solvers [22, 23] to sequential preparations of topological states [38–40]. The variational ansätze and random circuits all have a local depth that grows with the system size, which removes the only available trainability guarantee. The topological states cut along an extended seam fail because their interface grows with the length of the seam and the knitting cost grows exponentially with it. The families that remain admit an affordable cut, but most of them leave subcircuits that a classical computer simulates efficiently, making the advantage disappear. Only one of the tested families combines an affordable cut with branch tasks that are provably hard, and even there the hardness holds only in the worst case. These observations point to a more specific architectural requirement. A useful distributed circuit should allow its computational reach to grow with system size without increasing either the local depth that controls the available trainability guarantee or the interface that controls the knitting cost. Finite local-depth circuits (FLDCs) provide a natural way to separate these conditions [41]. Instead of the total depth, they bound the size of each local block and the number of blocks acting on each qubit. For blocks drawn independently from local 2-designs and local costs satisfy-

ing the coefficient condition of Sec. II, this keeps gradient variances bounded away from zero independently of the system size, even when the total depth grows linearly with n. FLDCs can therefore prepare long-range entangled states, and their local observables can depend on a region of the circuit that grows with the system size. If only a bounded number of gates connect the subcircuits, the knitting cost also stays bounded. Distributed FLDCs with bounded interfaces thus provide a candidate architecture in which gradient-resolution guarantees are compatible with large total depth, under the stated local-cost and initialization assumptions, and in which affordable knitting, trainability, and classical difficulty might coexist. Third, we test this candidate on the simplest bounded interface, two open patches of the field-perturbed toric code [42, 43] joined by one variational entangling gate, extending the benchmark of Ref. [41]. Cutting the bridge gate leaves two subcircuits which we run on an IBM Nighthawk processor and recombine classically [14, 15]. We simulate up to 48 spins and measure up to 142 on the device, with directly optimized angles up to 34 spins and, beyond, zero-field angles that prepare the exact ground state without the perturbing field. The hardware experiment is a classically verifiable test of whether an affordable bridge reconstruction recovers physically relevant information that running the two patches independently would miss. Our simulations confirm that the knitting cost does not grow with the size and that the energy gradients show no exponential decay. On the device, the knitted energies reproduce the field dependence of the ideal circuits, but their signal falls with

3 the depth of the routed subcircuits, by about 0.6% per layer of two-qubit gates, with no error mitigation beyond dynamical decoupling, twirling, and post-selection [44–46]. Since the bridge changes the energy per spin only at order 1/N , we also knit the connected correlation across the contact, which independent execution cannot produce. It stays clearly nonzero up to 98 spins, from about 87% of its target at 8 spins to about a fifth at 80 and 98 spins, so the reconstruction retains part of the cross-patch information within the limits of device noise. The remaining challenge is state preparation at scales beyond classical simulation. Resolvable gradients make training possible in principle with polynomially many shots, but exact classical simulation limits it in practice, which is why the larger patches use the zero-field angles. We propose a parameter transfer motivated by quasiadiabatic continuation, which carries angles from short strips to longer ones without further optimization, see App. G. We leave the extension to compact patches to future work. Section II states the three requirements and fixes the notation, Sec. III formulates the criteria and applies them, Sec. IV presents the toric-code test, and Sec. V discusses what the results imply. Figure 1 summarizes this structure, while the appendices collect all technical details. II.

REQUIREMENTS FOR DISTRIBUTED QUANTUM ADVANTAGE

the subcircuits separately, with the operations of branch α inserted, and knitting their outcomes classically. We call the estimation of local expectation values of one such subcircuit a branch task. Sampling branch α with probability pα = |cα |/Γcut and weighting its knitted outcome by Γcut sgn(cα ) gives an unbiased estimator of ⟨O⟩. For a Pauli-product observable measured in its product eigenbasis, branch outcomes lie in [−1, 1], so the weighted outcomes lie in [−Γcut , Γcut ]. Hoeffding’s inequality [47] then shows that M≥

i

i

see Fig. 2(b). The coefficients as,i are real, and each map Fs,i is implemented by local operations, measurements, and classical post-processing. For κ crossing gates decomposed independently, a branch α = (i1 , . . . , iκ ) replaces each crossing gate s by Fs,is and leaves all other gates unchanged, see Fig. 2(c). This defines a full-circuit branch Qκ map Fα with coefficient cα = s=1 as,is , and X X U= cα Fα , Γcut = |cα |, (2) α

α

where U(ρ) = U ρU † is the channel of the original circuit. By linearity, ⟨O⟩ is the same signed sum of branch contributions. Each contribution is obtained by running

(3)

independent shots suffice for additive error ε with failure probability at most δ, where a shot runs the subcircuits of one chosen branch once and knits their outcomes. The sampling cost therefore grows as Γ2cut . We call a family cheaply cuttable when Γcut = O(1) and efficiently cuttable when Γcut grows at most polynomially with n. Cut requires efficient cuttability. The value of Γcut depends on both the crossing gates and the chosen decomposition. We define the interface capacity κ(Π) as the number of two-qubit gate channels severed by Π, with repeated gates counted separately. For crossing gates of the form exp[−iθs (Ps ⊗ Qs )/2], with Ps and Qs single-qubit Pauli operators, optimal decompositions of the individual gates give [18] Γind =

This section states the requirements Cut, Sim, and Train for a circuit family as n grows. We consider the task of estimating expectation values ⟨O⟩ at the output U |0n ⟩ of an n-qubit circuit U , for observables O with support on at most r = O(1) qubits and norm ∥O∥ ≤ 1. Circuit cutting estimates full-circuit expectation values from measurements on smaller subcircuits. Fix a partition Π that assigns each qubit to a subcircuit, see Fig. 2(a). Each gate channel Vs crossing Π admits a signed quasiprobability decomposition into subcircuit-local operations [14, 16], X X Vs = as,i Fs,i , γs = |as,i |, (1)

2Γ2cut 2 ln ε2 δ

κ Y

1 + 2| sin θs | ≤ 3κ .

(4)

s=1

For parallel crossing gates, an optimal joint decomposition, which treats them collectively, achieves the smaller value [17, 18] Γjoint = 2

κ Y

1 + | sin θs | − 1 ≤ 2κ+1 − 1.

(5)

s=1

In both cases, a bounded interface keeps Γcut bounded however large the subcircuits become. Neither decomposition requires classical communication between subcircuits during execution, so the subcircuits can run on separate processors or one after another on the same processor, as in the local-operations setting of Ref. [15]. Entanglement across the partition also constrains Γcut , independently of the chosen decomposition. For a pure output |ψ⟩ and a bipartition Π = A:B, let λa be the eigenvalues of the reduced state ρA . We define the seam entropy as the Rényi-1/2 entropy X p  S1/2 (ρA ) = 2 log2 λa , (6) a

and use it to measure how much entanglement a reconstruction has to reproduce across the partition. Appendix A collects the bounds that relate it to Γcut . An efficient cut does not by itself imply an advantage. We call a branch task efficiently simulable when a classical

4 (a)

(b)

(c)

(d)

blocks 1 U1

A V1

V2

Π

Vs =

X i

B

as,i

A Fs,i

i1

i2

i1

i2

2 U2

cα

2

B Fs,i

U3 2 U4 1

FIG. 2. Circuit cutting turns crossing gates into sums of local operations, and finite local depth bounds the blocks acting on each qubit. (a) A partition Π assigns the qubits of a circuit to two subcircuits A and B. Gates acting within one subcircuit (gray) are untouched, and the gates Vs that cross Π (blue) are removed. (b) Each crossing gate is written as a signed sum of products of operations local to A and to B, with real coefficients as,i . (c) A branch α = (i1 , i2 ) chooses one term for every crossing gate (blue). Q The two subcircuits are then independent, and the original expectation value is the sum over branches weighted by cα = s as,is . (d) A finite local-depth circuit in the sense of Definition 1, here one sequential sweep of blocks Uk on neighboring spins. Its depth grows with the number of spins, but every block acts on β = 2 spins and every spin is acted on by at most χ = 2 blocks, as the counts on the right show.

algorithm achieves inverse-polynomial additive error in polynomial time. Sim requires that at least one branch task entering the knitting is not efficiently simulable. If every branch task is efficiently simulable, so is the knitted estimate (App. B). For a parametrized circuit U (θ), we take the cost P λ ⟨O ⟩ C(θ) = to be a weighted sum of expecν θ ν ν tation values of observables with support on at most r qubits, and write ∂µ C = ∂C/∂θµ . Train requires that Varθ [∂µ C] at random initialization decays at most polynomially with n, which excludes a barren plateau. For trainability, we rely on the guarantee available for finite local-depth circuits: Definition 1 (Finite local-depth circuit). For a parametrized n-qubit circuit decomposed into unitary blocks, U (θ) = UM (θM ) · · · U1 (θ1 ), the maximum block support and the local depth are β = max | supp(Uk )|, k

χ = max {k : q ∈ supp(Uk )} , q

where supp(Uk ) denotes the qubits acted on by block Uk , so χ is the maximum number of blocks acting on any one qubit, see Fig. 2(d). A family of such circuits is a finite local-depth circuit (FLDC) family if it admits decompositions for which β and χ remain bounded independently of n. We call a circuit an FLDC when it belongs to such a family. When the blocks are drawn independently from local 2-designs, Zhang, Liu, and Zhang show that Varθ [∂µ C] ≥ g [41]. The constant g depends only on r, β, χ, and the coefficients λν of the cost terms whose support overlaps the block containing θµ . It is therefore independent of n whenever the sum of the squares of these coefficients is bounded away from zero. The bound holds for any total depth and is stated in full in App. C.

III.

ASSESSING CIRCUIT FAMILIES FOR DISTRIBUTED EXECUTION

This section turns the three requirements into criteria that can be checked for a given circuit family, and then applies them to circuit families from the literature. Each criterion is a screen, not a certificate. Passing all of them identifies a candidate, not a demonstrated quantum advantage. A failure excludes a family only where the criterion states a necessary condition. A.

Assessment framework

The assessment takes two inputs. The first is a circuit generator, an algorithm that returns, in time polynomial in n, the gate list, qubit supports, observables, and parameters of the n-qubit circuit. A generator specifies a family, so the criteria can ask how resources scale with n rather than how a single instance behaves. The second input is a device model (w, D, M, ε, δ). It allows D subcircuits of at most w qubits each and a budget of M shots per expectation value, which is to be estimated to additive error ε with failure probability at most δ. An experimental budget must additionally account for the subcircuit executions behind each shot and for the number of observables. The partition is chosen as part of the assessment. For a fixed gate representation, the smallest interface over balanced partitions is κ∗ (n) =

min

Π balanced

κ(Π),

(7)

where balanced means that every subcircuit holds n/D qubits up to a fixed relative tolerance. Every candidate subcircuit must also fit within w qubits. Finding this minimum is computationally hard, so a heuristic search returns only an upper bound on κ∗ (n). A small bound exhibits an affordable partition. A large bound proves nothing, because a better partition may exist. We therefore report the partition that was found and never treat a failed search as an exclusion.

5 B.

Assessment criteria

Table I lists the eight criteria, grouped by the requirement they test. Each criterion belongs to one of three tiers, which record the kind of evidence it needs. Tier-1 criteria are structural checks computed from the generator alone. Tier-2 criteria are numerical diagnostics at sizes that a classical simulator can reach. Tier-3 criteria are theoretical obligations, met only by a proof or a citation and otherwise recorded as open. Open entries are kept separate from failures throughout. Knitting cost (Cut). Criterion C1 asks whether a partition can be knitted within the budget. Equation (3) gives a worst-case sample budget for the stated quasiprobability estimator, and we use its inverse as a conservative norm-budget threshold, s M Γmax = ε . (8) 2 ln(2/δ) C1 passes when Γcut ≤ Γmax , while a larger norm does not, by itself, exclude a task-specific or variance-aware estimator. Inserting the worst-case norms of Eqs. (4) and (5) turns this into a largest number of severed gates, for Γmax ≥ 1,   Γmax + 1 κjoint = log , 2 max 2 (9) ind κmax = ⌊log3 Γmax ⌋ . For the ancilla-free joint construction assessed here, the severed gates share a time slice. Both counts grow only logarithmically with M , and App. A 4 evaluates them for typical budgets. Each additional severed gate multiplies the worst-case shot count by nine for independent decompositions and by a factor approaching four for joint ones. When the gate angles are known, the actual norm replaces these worst-case counts. Criterion C1 answers only half of the practical question, because the subcircuits must also fit on the devices. For a fixed partitioning strategy, let ncut be the largest size at which knitting remains affordable and nwidth ≤ Dw the largest size at which the subcircuits fit. The family can then be run up to nmax = min(ncut , nwidth ). We call it width-limited when ncut ≥ nwidth , since the hardware sets the reach. Figure 3 shows both cases. A bounded interface whose norm is within budget is width-limited, while an interface whose norm grows with n eventually becomes cut-limited. Asymptotic and finite-size conclusions can differ. A constant knitting cost may still exceed the budget, and a growing interface may stay affordable over the sizes of interest. Criterion C2 gives a lower bound that no exact statereconstructing decomposition in the stated class can beat. It rests on the following identity, proved in App. A. Proposition 1 (Cost–entropy identity). For a pure target |ψ⟩ across A:B, the minimum coefficient one-norm of a

gates severed nwidth

√ κ∗ = Θ( n)

cut limited

width limited κmax ∗

κ = O(1) Dw

n

FIG. 3. A bounded interface leaves the affordable region only through the device width. The budget fixes a threshold κmax on the number of severed gates, independent of the circuit, while the circuit family fixes an interface curve κ∗ (n), independent of the budget. The shaded region is both affordable and within device width. A bounded interface leaves it at nwidth , so hardware sets the reach. A growing interface leaves it through the budget threshold first unless the device width is reached earlier. The curves show minimum interfaces, whereas a heuristic search supplies upper bounds, and known gate angles should be assessed with the actual norm.

signed decomposition into normalized separable states is Γsep (ψ) = 2 · 2S1/2 (ρA ) − 1.

(10)

Any exact quasiprobability knitting from a product input, whose branches use local operations and classical communication, satisfies Γcut ≥ Γsep (ψ). The first statement is the robustness of entanglement of a pure state [48], expressed through the seam entropy, and the second is its consequence for exact knitting. Lower bounds of this kind on the sampling overhead of quasiprobability decompositions are known for gates and wires [19, 20]. The version here is stated for the output state, through its seam entropy, and therefore holds for every decomposition of every gate set at once. The bound applies to one state and one partition. If Γsep > Γmax , no exact knitting protocol of this kind is affordable at that partition within the norm-budget model. Excluding a family requires the bound at every admissible partition, so a large entropy at a single partition is not enough. Conversely, a small Γsep does not provide an affordable protocol. The identity concerns exact reconstruction of the state, not every method that estimates a single observable. The interface limits the seam entropy but does not fix it. Starting from a product state, each severed two-qubit Pauli rotation at most doubles the Schmidt rank across the partition, so S1/2 (ρA ) ≤ κ whatever the subcircuitlocal gates are (Proposition 2 in App. A 2). The entropy actually reached can lie anywhere below this bound. C2 is therefore evaluated on the states the task requires, including the trained states of a variational circuit. Classical difficulty (Sim). An affordable cut matters

6 Criterion

Quantity or task

Tier

What a pass or a failure shows

Is the knitting affordable? (Cut) C1

interface capacity

κ(Π), bounds on κ∗ (n), and Γcut against Γmax

1

C2

seam entropy

S1/2 (ρA ) and the cost floor Γsep

2

A partition within budget establishes feasibility. A large heuristic cut does not show that every partition is unaffordable. A large floor excludes exact knitting at the assessed partition. A small floor does not certify affordability.

Does classical difficulty survive the cut? (Sim) S1

where hardness lives

Branch tasks of each subcircuit

3

S2

light cone

1

S3

task type

Backward causal cones of the required observables Task and accuracy covered by the supporting results

3

Efficient branch tasks together with efficient knitting exclude an advantage for the assessed task. Hardness must survive in the branch tasks. Bounded cones permit direct simulation. Growing cones do not establish hardness. Results must cover local expectation values. Hardness of sampling alone is insufficient.

Are gradients resolvable? (Train, variational families only) T1

local structure

T2

guarantee constant

T3

gradient variance

β and χ for a stated block decomposition Applicability, scaling, and magnitude of g Varθ [∂µ C] against n

1 1 2

Bounded values identify an FLDC. A failure is not a general untrainability result. A usable bound supports gradient resolution. A small or inapplicable bound leaves the variance undecided. Numerical scaling with uncertainty for a stated ensemble. Finite-size evidence is not an asymptotic guarantee.

TABLE I. The screening criteria. The tier records the kind of evidence a criterion needs. 1 structural checks computed from the circuit generator. 2 numerical diagnostics at sizes reachable by classical simulation. 3 theoretical obligations met by a proof or a citation. Outcomes refer to the specified task, partition, parameter regime, and resource budget. For non-variational families the Train criteria do not apply.

only if classical difficulty survives it. Criterion S1 rests on the following reduction. Theorem 1 (Efficient knitting preserves classical simulability). Let a bipartition sever κ = O(log n) gates, each replaced by at most m = O(1) efficiently specified local terms with uniformly bounded coefficient norm and no communication between subcircuits. If the branch tasks of both subcircuits are in BPP, uniformly over branches and to inverse-polynomial additive error, then estimating ⟨O⟩ for observables of bounded support is also in BPP. The proof, in Theorem 5 and Corollary 6 of App. B, expands the observable into products of Pauli operators on each subcircuit and estimates every branch separately. With κ = O(log n) there are at most mκ branches and Γcut is polynomial, so polynomially many branch estimates at inverse-polynomial precision suffice. An efficient cut whose branch tasks are classically easy is thus classically easy as a whole. Two consequences follow for S1. Simulating the subcircuits with the severed gates simply deleted does not settle it, because the branch tasks include the inserted local operations. Likewise, a hardness result supports S1 only if it concerns these branch tasks. Appendix D constructs a family in which such hardness sits inside one

subcircuit of a bounded-interface cut. The observation behind the theorem, that knitting classically simulable subcircuits yields a classically simulable estimate, is made in Ref. [19], and the hybrid simulation of Ref. [11] uses the same expansion with a cost exponential in the number of severed gates. Theorem 1 states it with the branch tasks as the unit of hardness, which is what criterion S1 needs. Criteria S2 and S3 determine how simulation evidence is read. S2 records the backward causal cones of the required observables, that is, the gates that can influence them and the qubits these gates act on, including the local operations inserted by the cutting branches. Uniformly bounded cones permit direct classical simulation, whereas growing cones do not establish hardness (Proposition 4). The generator-level diagnostic in App. E reports the largest probed gate-list cone, whose interpretation depends on the stated circuit representation and probe set. S3 checks that a cited simulation or hardness result concerns local expectation values at the required accuracy. Hardness of sampling alone does not meet it. Together, S2 and S3 separate evidence against an advantage from the mere absence of a known simulation. Gradient resolution (Train). For variational families,

7 criterion T1 records β and χ for an explicit block decomposition and checks whether the family is an FLDC. T2 records whether the bound g of Sec. II applies, how it scales, and how large it is. A size-independent g can still be too small to help at a given budget. Neither a failed T1 nor a small g establishes a barren plateau, and Proposition 6 gives a sufficient condition for one. T3 therefore measures the gradient variance directly over a range of sizes, for a stated initialization ensemble and cost normalization. A variance that does not decay supports trainability only over the tested range. Random initialization and a warm start answer different questions, and T3 concerns the former. In practice, one first fixes the task and budget, searches for partitions, and runs the Tier-1 checks. Tier-2 diagnostics then refine the families that survive, and the Tier-3 obligations decide which claims can be made. Finally, the reachable sizes must be compared with the sizes at which classical methods still perform the same branch tasks at matched accuracy.

C.

Scaling regimes

The criteria also constrain how a computation may be distributed. Growing the subcircuits and growing their number, the two ways of scaling a distributed computation, have opposite consequences. Theorem 2 (Scaling regimes for distributed circuit cutting). With D = O(1) and w = Θ(n), an interface capacity κ(Π) = O(1) gives Γcut = O(1), and κ(Π) = O(log n) gives Γcut = poly(n). With w = O(1) and D = Θ(n/w), a connected arrangement severs at least D − 1 gates. If | sin θs | is bounded below by a positive constant on a constant fraction of them, independent cutting has Γcut = 2Ω(D) , and the joint decomposition has the same scaling whenever its time-slice condition holds. √ Independently, a pure target with S1/2 (ρA ) = Ω( n) across a √ balanced grouping of the subcircuits obeys Γcut = 2Ω( n) for the exact knitting protocols of Proposition 1. Theorem 2 is proved in App. A 4. Its first two statements concern the norms of the stated constructions and need not bound the best decomposition of a particular known circuit, whereas the entropy statement holds for every exact protocol in the stated class. For the cutting constructions and target-state assumptions of the theorem, growing the subcircuits behind a bounded or logarithmically growing interface permits efficient reconstruction, with a constant or polynomial knitting cost, whereas the fixed-width regime incurs an exponential overhead. Many small subcircuits would stay affordable only if the problem itself split into many weakly coupled pieces, which suggests that such a problem is close to classically parallel from the outset. A cut can exploit a bottleneck that the physics provides, but it cannot create one. Even in the favorable regime, a bounded interface

supplies neither hardness nor trainability, and both must be established separately.

D.

Assessment of existing circuit families

Table II applies the criteria to eighteen circuit families and regimes from the literature. Each column corresponds to one criterion of Table I, ordered by tier, and each row records the outcome in a specified regime. Appendix E reports the Tier-1 criteria for seven explicit generators associated with these families and a constant-depth control, over sizes from 7 to 517 qubits (Fig. 8). Four patterns emerge among the assessed families. Most assessed families lack one or more structural guarantees. Every conventional variational circuit in the table (rows 1–6) has a local depth χ that grows with system size, through repeated ansatz layers or, for QAOA, through the swaps added by routing on a planar device. For ADAPTVQE with the assessed generalized operator pool (row 4), the block support β grows as well; the verdict concerns this FLDC screen, not all adaptive variants [56, 57]. The bounded-cone statement for row 5 applies only at fixed p and bounded logical degree, not to dense graphs or the conservative routed gate-list cone of App. E. Where evaluated, the formal bound g is strongly suppressed, reaching values as small as 10−2167 for row 4. This makes the FLDC bound uninformative at the assessed sizes, since it does not establish a barren plateau. For the hardware-efficient ansatz and deep scrambling circuits (rows 3 and 6a), gradient suppression is supported separately by the T3 assessment. Several assessed constructions have an interface that grows with the linear size L of the partition boundary (rows 8b, 9, 10, 11a, 12a). The S3 example in row 9 is distinct from the related D4 preparation settings of Refs. [74, 75]; its fixed-state assessment does not cover fusion-assisted universal computation. For them, knitting is exponentially expensive in L. Affordable cutting is typically associated with classically tractable subcircuits. Sequential matrix-productstate preparation (row 7), the Clifford toric code with a bounded contact (row 8a), and strongly injective isometric tensor-network states (row 11b) all admit a constant interface. Their subcircuits, however, are classically tractable, and by Theorem 1 so is the knitted computation. Row 7 refers to the sequential unitary construction; logarithmicdepth unitary [68] and adaptive constant-depth [69] preparations are alternative resource models. For the assessed tasks, efficient classical simulation of the branch contributions also makes the reconstructed expectation values classically tractable. The easy verdict for the Hubbard ladder (row 1b) concerns the groundstate reference calculation, while efficient simulation of all required variational branches would need a separate argument. Few assessed families exhibit both classical hardness and affordable cutting. The families with the strongest hardness evidence carry seam entropy that grows with sys-

8 TABLE II. Circuit families and regimes assessed against the criteria of Table I. Tier-2 entries are evidence at the sizes tested, not proof. Daggered entries are subject to limitations of the specific task considered, and are discussed in Sec. III D. ✓ passes the stated criterion, ✗ fails it, ? is unresolved, and – is not applicable. For non-variational families, (β, χ) are reported as structural descriptors rather than as a trainability assessment. Highlights mark promising cases: bounded-interface theoretical candidate , application studied here . Tier numbers as in Table I. 1 structural

No. Circuit family/regime (β, χ)

g

T1

T2

2 numerical 3 theoretical

cone κ∗ (n) S2

C1

Var

S1/2

hard

task

T3

C2

S1

S3

Verdict

A. Conventional variational circuits and scrambling benchmarks Hubbard variational circuits: 1a (a) EHV ansatz [49–51] (2, Θ(n)) 1b (b) Hamiltonian variational ansatz on a Hubbard ladder [51, 52] 2

J1 –J2 Heisenberg ground-state ansatz [53]

3

Hardware-efficient ansatz for molecular VQE [23, 26, 31, 54]

4

ADAPT-VQE with a generalized operator pool [55–57]

≤10−43

✓

?

?

?

✗

✓

No FLDC guarantee. Preparation and seam both grow.

(2, ω(1))

≤10−168

✓

ω(1)

?

O(1)

✗

✓

Low seam entropy, but classically easy: DMRG solves the ladder.†

√ (2, Θ( n))

≤10−86

✓ Θ( n)

?

√ Θ( n)

?

✓

No trainability guarantee, and not cuttable.

(2, Θ(d))

≤10−28

✓

ω(1)

✗

?

✗

✓

Gradient suppression in the assessed global-cost regime.

(Θ(n), ω(1)) ≤10−2167

✓

ω(1)

?

?

✗

✓

No FLDC guarantee. Tested molecules are exactly solvable.

4−ω(1)

?

Θ(n)

?

?

?

✓

Growing local depth; branch complexity unresolved.

4−ω(1)

✓ Θ( n)

✗

Θ(n)

✓

✗

Hard only to sample, not for expectation values. Volume-law seam, not cuttable.†

✓

✓

?

✓

✗

Hard only to sample, and the interface grows.†

5

Routed or dense (2, ω(1)) QAOA [58–62] Circuits hard only to sample: 6a (a) deep or scrambling (2, ω(1)) circuits [31, 32, 63–65]

6b (b) fermionic brick-wall with k particles [36]

(2, Θ(n))

–

√

√

Θ(n)

B. Sequential preparation and calibration examples (log D, O(1)) D−O(1)

7

Sequential MPS preparation [9, 66–69] Clifford toric code: 8a (a) bounded contact [42, 70, 71]

8b (b) Extensive seam [38, 42, 72] 9 Sequential S3 quantum double (non-Abelian) [73–75] 10 Fibonacci string net (universal braiding) [71, 76]

✓

O(1)

✓

O(1)

✗

✓

Cuttable, no advantage.

(2, ≤6)

–

✓

O(1)

–

O(1)

✗

✓

Cuttable, no advantage.

(2, ≤6)

–

✓

Θ(L)

–

Θ(L)

✗

✓

(6, O(1))

–

✓

Θ(L)

–

Θ(L)

✗

✓

Clifford, so classically easy. Calibrates seam cost 2L − 1. Classically easy: solvable group, braiding not universal. Seam cost 6L−1 .†

(5, O(1))

–

✓

Θ(L)

–

0.92L

✗

✓

Classically easy state (braids of length ≤3). Seam cost ≈1.9L .†

C. Two-dimensional FLDCs with contrasting interfaces and simulability Sequential isoTNS: 11a (a) transparent seam [39, 40, 77, 78]

(log D, O(1)) D−O(1)

✓

Θ(L)

✓

√ Θ( n)

✓

✓

Hard, but√not cuttable: seam entropy grows as n.

11b (b) Strongly injective (η ≥ 0.41), bounded interface [77, 79, 80]

(log D, O(1)) D−O(1)

✓

O(1)

?

O(1)

✗

✓

Cuttable, no advantage. Noise threatens T3.

11c (c) Weakly injective, bounded interface [39, 40, 77]

(log D, O(1)) D−O(1)

✓

O(1)

?

≤κ

✓

✓

Candidate. Worst-case hardness; T3 unresolved.

(2, ∼6)

∼10−29

✓

Θ(L)

✓

L−1

?

✓

One code cut along its boundary, unlike 12b. Cost 2L − 1, so L ≤ 6.

(2, O(1))

–

✓

1

✓

≤1

?

✓

Cheap cut, but S1 open.

Variational toric code: 12a (a) transparent seam [38, 41, 81–84] 12b (b) Field-perturbed, single bridge (this work)

9 tem size. Sequential isometric tensor-network states with a transparent seam (row 11a) are hard √ in the worst case, but their seam entropy grows as n, so no exact knitting protocol is affordable (Proposition 1). The hardness ticks in rows 6a and 6b refer to sampling evidence, not a verified S1 result for local-observable branches. The Paulipropagation guarantee of Ref. [32] is ensemble dependent and polynomial at fixed error, becoming quasi-polynomial at inverse-polynomial accuracy; it does not by itself supply the uniform branch simulator required by S1. Weakly injective isometric tensor-network states with a bounded interface (row 11c) provide a candidate with worst-case hardness; numerical trainability for a specified initialization ensemble remains a separate assessment. Corollary 7 in App. D shows that such families exist, but it does not identify natural instances. The injectivity-dependent local-observable results are those of Ref. [77, Sec. V], with η = δ 2 D2 in that source; Refs. [79, 80] provide related circuit and variational context. Topological state families provide a benchmark for seam cost. For the Clifford toric code with an extended seam (row 8b), the knitting cost is known and grows exponentially with the seam length (Theorem 4 and Corollary 3 in App. A 3), which follows from the analytic seam entropy L − 1 of its fixed-point stabilizer state. The non-Abelian examples (rows 9 and 10) show that richer anyon content does not change this picture. The easy verdicts in rows 9 and 10 are restricted to the stated fixed-state or short-braid benchmarks, while neither nonuniversal braiding nor a preparation reference alone establishes efficient simulation of arbitrary cutting branches. The variational toric code with a transparent seam (row 12a) inherits this exponential seam cost when its states retain enough of the fixed-point seam correlations (Corollary 5), and at the fixed point the cost 2L − 1 fits the norm threshold of Eq. (8) only up to L ≤ 6 for M = 109 , ε = 0.01, and δ = 0.05. Our single-bridge application (row 12b) severs one gate, so its Schmidt rank across the seam is at most two and its knitting norm 1 + 2| sin ϕ| never exceeds three. Its local depth is bounded, and its sweeps are applied sequentially. The guarantee constant does not apply because the fixed encoder, the bridge, and the anchor are not drawn from the local 2-designs that the bound of Sec. II assumes. We therefore measure the gradient variance of criterion T3 directly (Sec. IV C). No single criterion decides the question. A cheap seam does not imply an advantage, while a large seam entropy at every admissible partition excludes exact knitting regardless of the decomposition. Passing every structural and numerical check still leaves the theoretical obligations open, and it is these obligations, above all S1, that decide whether distribution can deliver an advantage.

IV.

A SINGLE-BRIDGE DISTRIBUTED GROUND-STATE EXPERIMENT

The screen of Sec. III identifies bounded-interface FLDCs as a candidate class in which affordable knitting, trainability, and classical difficulty can coexist. Its Tier-1 criteria follow from the circuit generator alone. The Tier-2 and Tier-3 criteria depend on the states that a circuit prepares and on its branch tasks, so they can only be decided for a concrete model. This section tests them on the simplest nontrivial interface, two FLDCs joined by one entangling gate (row 12b of Table II). The bounded interface controls the knitting cost, but it does not determine the quality of the variational state, the resolution of its gradients, or the classical difficulty of its branch tasks. Because the branch tasks remain classically tractable at the sizes we run, the device results can be compared with classical references, which makes this a verifiable test of what bridge reconstruction recovers beyond independent execution. Section IV A fixes the model and the circuit (App. F). Section IV B evaluates the circuit and the ground state classically and prepares the angles (App. G). Section IV C checks the knitting cost and the Tier-2 criteria in simulation (Apps. H and I). Section IV D reports the hardware results (App. J). Throughout this section, n denotes the number of spins in one patch and N = 2n the total number of spins. A.

Model and variational circuit

We consider two patches A and B of Kitaev’s toric code in a parallel magnetic field 0 ≤ h ≤ 1 [42, 43, 82, 83], connected by a contact of strength t = 0.5 between the facing exterior spins a ∈ A and b ∈ B,  X X X H(h, t) = − (1 − h) Sv,p + Pf,p v

p∈{A,B}

−h

N −1 X

f

(11)

Xj + Zj − t Za Zb ,

j=0

where, for every vertex v and face f of patch p ∈ {A, B} and with X and Z the Pauli matrices, the star and plaquette operators are Y Y Sv,p = Xe,p , Pf,p = Ze,p . (12) e∋v

e∈∂f

Figure 4(a) shows one of each. A star on the boundary acts on its two or three edges. A patch is a rectangle of nx × ny square plaquettes with one spin on every edge, so that the number of spins in one patch is n = 2nx ny + nx + ny .

(13)

Our task is to prepare the ground state of Eq. (11) variationally and to estimate its energy. We use the toric

10 (a)

(b) |0⟩A

Sv

a

VApre

EA

RSv (α) VApost

MX /MZ

−tZa Zb a

RZZ (ϕ)

b Pf

HA (h)

|0⟩B

VBpre

EB

VBpost

b

MX /MZ

RZZ (ϑ5 )

HB (h)

4

3

4

3

1

2

1

2

RY (ϑ1 ) RZ (ϑ3 )

(c) a

b

MZ

Z RZZ (ϕ)

=

c2

+ s2 Z

R+

R+

MZ − d

+ d

R−

R− − d

+ d MZ

MZ

FIG. 4. Two toric-code patches joined by one variational gate, the sweeps that prepare their states, and the six branches that knit them. (a) Two open toric-code patches with spins on the edges (dots). A star Sv and a plaquette Pf are highlighted, and the only cross-patch term of the Hamiltonian couples the facing exterior spins a and b. (b) Circuit order. Fixed encoders Ep prepare the zero-field ground states, the prefix sweeps Vppre act before the bridge RZZ (ϕ), and the anchor rotation RSv (α) and the suffix sweeps act after it. The inset shows one sweep on two neighboring faces. In each face it applies a pair block to each of the four pairs of consecutive edges (arcs) in the order of the numbers, and it visits the faces row by row (arrow), so a spin on an edge shared by two faces (ring) enters four pair blocks per sweep. A pair block rotates both spins by RY and RZ and couples them by RZZ , Eq. (F2). The anchor rotation and the suffix sweeps of patch A together form VApost of Eq. (15), so the anchor is applied once. Each patch is read out in the all-X or the all-Z basis. (c) The bridge as a weighted sum of products of local operations on spin a (top wire) and spin b (bottom wire), read as maps on density matrices. Here c = cos(ϕ/2), s = sin(ϕ/2), d = 12 sin ϕ, and R± = RZ (±π/2). The terms of weight c2 and s2 apply the identity or Z on both spins, and the four terms of weight ±d come from the commutator. A Z measurement MZ lets the circuit continue and multiplies the outcome of that shot by the recorded sign. The two patches never interact.

code [42] because its zero-field, i.e. h = 0, ground state needs exactly what FLDCs offer, since no constant-depth local circuit prepares it [85, 86], whereas a sequential circuit of bounded local depth does [38, 87]. The system is solvable at zero field, where the ground state is the common stabilizer state of the two patches mixed by the contact with a single excited state, and at h = 1, where it is a product state apart from the contact pair (Apps. F 2 and F 4). In between, the field and stabilizer terms do not commute and no closed form is known. The field drives the bulk out of the topological phase through a transition estimated between h ≈ 0.25 and h ≈ 0.29 [43, 82, 83], where the gap γN (h) between the two lowest levels is expected to become small. This region is the natural target of a variational study at large sizes, since classical methods are also expected to struggle there. Tensornetwork evaluations of the circuit and its branch tasks stop converging within our budget at the hardware sizes near and above the transition (Sec. IV B). Criterion S1 therefore remains open. The variational circuit mirrors the structure of the Hamiltonian [41], see Fig. 4(b). A fixed encoder Ep of Hadamard and CNOT gates prepares the zero-field ground state of each patch [87], and local sweeps then dress this state toward the ground state at h > 0 (App. F 1). A sweep proceeds through the faces row by row, and within each face it applies a pair block, made of RY and RZ

rotations on two adjacent spins and one RZZ rotation between them, to each of the four pairs of consecutive edges in a fixed cyclic order, see inset of Fig. 4(b). A single gate generated by the contact term couples the patches, as in the Hamiltonian variational ansatz [52],   ab RZZ (ϕ) = exp −iϕZa Zb /2 . (14) It is followed on patch A by an anchor rotation RSv (α) = e−iαSv /2 , generated by a star Sv incident to a, and on both patches by further sweeps. Collecting the sweeps into Vppre and Vppost , with VApost including the anchor rotation, and writing Ep for the encoders, the prepared state is
ab |ψ(θ)⟩ = VApost ⊗ VBpost RZZ (ϕ) (15)
pre pre × VA EA ⊗ VB EB |0N ⟩. We use two prefix and two suffix sweeps. A second sweep lets every pair block act again after all its neighbors have been dressed, which a single sequential sweep cannot provide, while the local depth stays bounded. Each spin enters at most four pair blocks per sweep and a bounded number of encoder, anchor, and bridge gates, so the circuit is an FLDC in the sense of Definition 1. ab We reconstruct the inter-patch bridge RZZ (ϕ) using the six-branch decomposition of Ref. [14], illustrated in Fig. 4(c). Each branch involves only patch-local operations, and the target expectation value is recovered as a weighted sum of the branch results.

11 Writing c = cos(ϕ/2), s = sin(ϕ/2), and d = cs = 1 2 sin ϕ, two branches apply either the identity or a Pauli Z 2 2

to both bridge spins, with weights c and s , respectively. The remaining four branches pair a signed Z measurement on one bridge spin with a rotation RZ (±π/2) on the other. For either choice of measured spin, the positive and negative rotations carry weights d and −d, respectively. A signed measurement means that the circuit continues after the measurement and the corresponding shot contribution is multiplied by the recorded eigenvalue m = ±1. The absolute branch weights sum to the knitting norm Γ(ϕ) = c2 + s2 + 4|d| = 1 + 2| sin ϕ|,

(16)

which depends only on the bridge angle and is independent of the patch size. App. H 1 combines the branches into the knitted estimator.

(App. H 1). Our noiseless simulations of the subcircuits and the knitting protocol reproduce the exact energies of the full system within about one error bar. Gradients are obtained from the exact evaluator by reverse-mode differentiation [90]. This evaluator is used both to train the angles and to compute the gradient variances of criterion T3 up to N = 48. Since the MPS evaluator does not provide reliable gradients at larger sizes within our computational budget, training is restricted to N ≤ 48. The larger patches measured on the device therefore use the zero-field angles, which are exact at h = 0 and require no training (App. F 3). A parameter transfer that supplies trained angles to longer strips without optimization at the target size is described in App. G, but not used in the hardware runs. C.

B.

Classical evaluation

This section describes the classical methods we use to estimate reference ground energies, the energies of the prepared circuits, and the gradients that train them. We also perform knitted simulations and check the Tier-2 criteria before the device runs. Because the bridge is RZZ (ϕ) = c 11 − is Za Zb , the prepared state splits into two product states, |ψ⟩ = c |A0 ⟩|B0 ⟩ − is |A1 ⟩|B1 ⟩,

(17)

where |p1 ⟩ is the state of patch p with a Z inserted on its bridge spin and |p0 ⟩ the state without it. The energy and its gradient therefore follow from quantities computed on one patch at a time, which lowers the cost from 2N to 2n . Up to N = 48, we evaluate the circuit energy Eθ = ⟨ψ(θ)|H|ψ(θ)⟩ exactly, with state vectors of one patch. Beyond that we use a matrix-product-state (MPS) evaluator at bond dimension 64, which matches the exact energies at N = 48 to about 10−3 per spin but does not reach this accuracy within our budget at N = 80 for h ≥ 0.25 and at N = 120. Its cost grows polynomially with the bond dimension, but the bond dimension needed for a given accuracy grows with the width of a patch [88], making the method not suitable for larger compact patches. For the ground energy, E0 , we use the closed forms at h = 0 and h = 1 for all N , and exact diagonalization at every field up to N = 24. Beyond that, we use the density-matrix renormalization group (DMRG) [89] at bond dimension 128, whose energies change by at most 10−4 per spin between bond dimensions 64 and 128 at every field up to N = 142. The energy is knitted from the six branches of Sec. IV A, shown in Fig. 4(c). Each subcircuit is read out in the all-X and the all-Z basis, which together cover every term of Eq. (11). The six branches, two patches, and two bases give 24 subcircuit settings. Each energy is estimated from M = 24,576 shots in total, split among the 24 settings so that subcircuits with a larger variance receive more shots

Simulated checks of Tier-2 criteria

Knitting cost (C1). In noiseless simulations of the protocol, the statistical error of the knitted energy per spin at fixed M does not grow with N at any field, and for h > 0 it even decreases (Fig. 5(a)). This is because the knitting norm of the single bridge, Eq. (16), stays at most three whatever the patch size, while the energy per spin averages over more terms as the patches grow. The trained bridge angles give Γ(ϕ) between 1.009 and 1.535, well below the worst case of three (trained angles of App. J 1). The number of shots needed for a given accuracy per spin therefore does not increase with the size, so C1 holds. Gradient variance (T3). We compute the variance of the gradient of the total energy at h = 0.25 over uniformly random angles with the exact evaluator, following Ref. [41], for strips of 8 to 38 spins and compact patches of 24 to 48 spins1 . The variance falls with N but not exponentially over these sizes (Fig. 5(b)), which is an improvement over circuits whose local depths grow with the patch and whose variances fall by a fifth with every added spin (App. I). The remaining decrease is a boundary effect, due to the gates near the open ends of a strip, which have a larger variance than gates in the bulk. Thus, longer strips, which add only bulk gates, have an average variance that levels off with increasing N , whereas compact patches, which have a larger share of bulk gates, have a lower variance that levels off only at larger sizes (App. I). Over the simulated sizes the gradients therefore remain resolvable with polynomially many shots, so T3 holds. Seam entropy (C2). Criterion C2 asks whether the entanglement across the seam stays small as the system grows, since by Proposition 1 it sets a floor on the knitting cost. We evaluate it for the ground state, which is what a

1 Strips have plaquettes dimensions 1 × k with k = 1, . . . , 6, giving

N = 8, 14, 20, 26, 32, and 38, and compact patches have 2 × 2, 2 × 3, 2 × 4, and 3 × 3 plaquettes, giving N = 24, 34, 44, and 48.

12

C1: knitting cost

(b)

2.5

h=0 h = 0:1 h = 0:2

10¡1

h = 0:25 h = 0:3 h = 0:5 h=1

Varµ [∂¹ E]

p

SE(E=N) M

2.0 1.5 1.0

(c) 1.0

T3: gradient variance FLDC strips FLDC compact patches

seam entropy S1=2 [bits]

(a)

10¡2

0.5 0.0

8

14

20

26

34

10¡3

10

20

N

30 N

40

50

C2: seam entropy capacity of one bridge N=8 N = 26 N = 14 N = 32 N = 20 N = 34

0.8 0.6 0.4 0.2 0.0

0.0

0.5

1.0

h

FIG. 5. In simulation, the single-bridge application passes the knitting and gradient checks. One√criterion per panel, from classical simulations of the ideal circuits. (a) Knitting cost (C1), the bootstrap error of E/N times M against N , one shade per field, from exact finite-shot simulations with M = 24,576 shots per point, with strips of 1 × k plaquettes joined by lines and the compact 2 × 2 and 2 × 3 patches as squares. (b) Variance of the energy gradient at random initialization (T3) against N , from exact simulation at h = 0.25 over 64 to 2,000 random draws of the angles, for strips (circles) and compact patches (squares). Points are medians over three groups of parameters sorted by causal distance, with the bridge angle excluded, and bands span the envelope of their 95% bootstrap intervals (App. I). (c) Seam entropy (C2) of the prepared states against the field, one shade per size, with bars of one standard error, the exact ground state for N ≤ 24 (gray), and the one bit that a single bridge can carry (dotted). Classical difficulty of the branch tasks (S1) has no panel and remains open.

single bridge has to support, and for the prepared states, which are what we knit. For the ground state, we evaluate Eq. (6) on its Schmidt spectrum from exact diagonalization, for N ≤ 24. Its seam entropy stays below about 0.2 bits, and almost all of its weight lies on its two largest Schmidt components, which mirrors the single bridge construction. For the prepared states, the seam entropy follows from the bridge angle ϕ and the expectation values za and zb of Za and Zb just before the bridge (App. H 3), q h i S1/2 (ρA ) = log2 1 + | sin ϕ| (1 − za2 )(1 − zb2 ) .

(18)

The prepared states also use at most about a fifth of the one bit that Proposition 2 allows for a single bridge, with almost no dependence on N (Fig. 5(c)), so C2 holds.

D.

Hardware setup and results

We run the knitted circuits on the IBM Nighthawk processor ibm phoenix for two sets of sizes, with the angles and the knitting protocol of Sec. IV B. The first set, N = 8, 14, 20, 24, 26, 32, and 34, uses the trained angles, and the second, the 3 × 3 to 6 × 6 patches with N = 48, 62, 80, 98, 120, and 142, uses the zero-field angles at every field. Every size is measured at h ∈ {0, 0.1, 0.2, 0.25, 0.3, 0.5, 1} and t = 0.5. In the four branches that measure a bridge spin, the measurement is deferred onto one ancilla, so

every subcircuit has n + 1 qubits, from 5 to 85, with 38 to 2,995 two-qubit gates. To suppress errors, we protect idle qubits against decoherence through dynamical decoupling sequences [44], and two-qubit gates through Pauli twirling [45], which turns coherent gate errors into stochastic Pauli errors. Additionally, we also post-select shots using the inversion test of Ref. [46], which discards shots with readout or relaxation errors at the end of the circuit (App. J 2). Post-selection keeps most shots at N = 8 but only a few percent at the largest sizes, so the number of shots that enter an estimate decreases with the size. No further error mitigation is applied to the raw energies. The hardware runs are designed as a classically verifiable test of whether an affordable bridge reconstruction recovers physically relevant information that running the two patches independently would miss. While the measured energies assess how accurately the device reproduces the prescribed coupled circuit, it is the connected contact correlation that tests the information that a bridge-off product state lacks. The experiment is thus a proof of principle for the value and the limits of bounded-interface reconstruction, not a demonstration of quantum advantage or a faster ground-state solver. Energies. On the device, the knitted energies reproduce the field dependence of the ideal circuits at every size (Fig. 6(a,b)). For the trained angles, they follow the ideal curves, including their maximum near the transition and their ordering by size, and for the zero-field angles they follow the exact linear dependence on h (Eq. (F5)).

13

(a) ¡0.4

258 CZ

¡0.6

E=N

¡0.8 ¡1.0

(b) 0.0

2,424 CZ

¡0.5

382 CZ

¡1.0

¡1.2

38 CZ

¡1.4 ¡1.6 0.0

N = 8 (1 £ 1) N = 14 (1 £ 2) N = 20 (1 £ 3)

0.2

0.4

N = 26 (1 £ 4) N = 32 (1 £ 5) N = 34 (2 £ 3)

h

0.6

0.0

¡1.5 ¡2.0

0.8

1.0

¡2.5

0.0

¡0.1 0.00 0.15 N = 48 (3 £ 3) N = 98 (4 £ 5) N = 62 (3 £ 4) N = 120 (5 £ 5) N = 80 (4 £ 4) N = 142 (5 £ 6)

0.1

0.2

h

0.3

0.4

h 0.30

0.5

FIG. 6. Device energies keep the shape of the reference at small sizes and lose most of their signal at large sizes. Energy per spin E/N against the field h, measured on the IBM Nighthawk processor ibm phoenix by knitting the bridge gate, one shade per size from light to dark. (a) Patches of N = 8 to 34 spins with directly optimized angles. (b) Patches of N = 48 to 142 spins with the zero-field angles, for h ≤ 0.5, with an inset enlarging the two largest patches. Markers joined by solid lines are raw device values, and dashed lines the ground-state energy, from exact diagonalization for N ≤ 24 and from the density-matrix renormalization group (DMRG) otherwise. Labels give the largest number of native two-qubit (CZ) gates in one routed subcircuit, and gray lines connect the energy axes of the two panels. Each point is one run of M = 24,576 shots, and error bars, where larger than the markers, are one bootstrap standard error from shot noise (App. H 1), computed from the post-selected records (App. J 2).

The magnitudes, however, are reduced by noise. For the trained angles, the ideal energies lie within 2 × 10−3 per spin of the ground-state reference, so the preparation is accurate and the reduction comes from the device. At h = 0, the fraction of the energy signal that survives falls from about 0.8 at N = 8 to about 0.04 at N = 142, steadily with the routed two-qubit depth of the subcircuits (App. J 3). A supplementary rescaling calibrated at h = 0 is assessed separately in App. J 4, and it is not applied to the raw energies of Fig. 6. The energy is, however, a poor measure of what the bridge contributes, since switching the bridge off changes the energy per spin by only 0.031/N at zero field, far below the device error (App. F 2). The reconstructed dependence of the energy on the bridge angle shows the same limitation (App. H 2). Contact correlation. We therefore measure the connected contact correlation of the final state Cab = ⟨Za Zb ⟩ − ⟨Za ⟩⟨Zb ⟩,

(19)

which vanishes for any product of patch states. Independent execution thus predicts Cab = 0 whatever the local parameters, and a nonzero value reveals correlations only the bridge can create. The energy per spin barely reflects it, since Cab changes it by only −tCab /N . We knit its three terms from the same branch measurements and combine √ them into Cab (App. H 1). At zero field, its target is t/ 16 + t2 ≈ 0.124 at every size (App. F 2).

On the device, knitting recovers a clearly nonzero contact correlation at every size up to N = 98, which independent execution would miss entirely (Fig. 7). At N = 8 the central value reaches about 87% of the target, so knitting removes most of the error of independent execution. The recovered fraction declines as the subcircuits grow deeper, to about 20% at N = 80 and 98, broadly following the loss of energy signal, and from N = 120 on the correlation is no longer distinguishable from zero. Bounded-interface reconstruction thus recovers a measurable part of the contact information on hardware, within limits that device noise tightens as the subcircuits grow.

V.

DISCUSSION AND OUTLOOK

Assessed jointly, affordable knitting, classical difficulty of the branch tasks, and resolvable gradients narrow where a distributed quantum advantage can be sought. An efficient cut whose branch tasks are classically easy is classically easy as a whole (Theorem 1), so a low knitting cost is never sufficient on its own. For the cutting constructions and target states we consider, knitting stays affordable only when the subcircuits grow behind a bounded or logarithmically growing interface (Theorem 2), since every severed gate multiplies the shot count and even a generous budget pays for only a handful of them. Among the eighteen assessed families, only one combines such a

14

Contact correlation Cab

0.125 Direct at h = 0 Fixed zero-field Analytic, t = 0:5 Bridge off: Cab = 0

0.100 0.075 0.050 0.025 0.000

¡0.025 0

50

100

150

N FIG. 7. Knitting recovers part of the contact correlation that independent execution misses. Connected contact correlation Cab of Eq. (19) at h = 0, knitted from the hardware records of Fig. 6, for the trained angles (circles) and the zero-field √ angles (squares). The dashed line is the target t/ 16 + t2 , and the solid line the bridge-off prediction Cab = 0. Error bars are 95% bootstrap intervals from shot noise, with the three expectation values resampled jointly.

cut with a hardness result for its branch tasks, and that result holds in the worst case. The outcomes of Sec. III fall into the four classes of Table III. Local depth, interface size, and branch-task complexity require separate assessment. A bounded interface controls reconstruction overhead, whereas finite local depth supports a gradient-variance bound only under the stated cost and initialization assumptions. Extensive causal cones do not establish hardness, as the classically simulable examples illustrate. We tested the bounded-interface FLDC in its simplest form, two toric-code patches joined by a single variational gate and executed as separate subcircuits. The experiment verifies what the bridge reconstruction recovers beyond running the patches independently, with a classical reference for every result. This makes it a testbed for practical distributed quantum advantage, since it checks the necessary ingredient that correlations across the cut survive the reconstruction as the subcircuits grow. The results show that the favorable geometry behaves as the criteria predict. In simulation, the single bridge keeps the knitting cost independent of the size, the prepared states use only a small part of the entanglement it can carry, and the gradients remain resolvable. On the device, knitting recovers a contact correlation that independent execution misses up to 98 spins, and the energies reproduce the field dependence of the ideal circuits up to 142 spins, with a signal that noise attenuates as the routed subcircuits grow deeper. Moving from these observations toward a distributed

advantage requires addressing three aspects. The first asks whether the circuits one would actually run are hard to simulate classically. Our application has bounded seam and local depth, but the hardness of its branch tasks is not established. Tensor-network calculations that stop converging at the hardware sizes are no evidence of hardness, and for bounded-interface subcircuits hardness is known only in the worst case over instances engineered to be hard (Theorem 7). What is still missing is a physically motivated Hamiltonian whose bounded-interface circuits keep their branch tasks hard. The second asks how many processors a single computation can usefully be split across. Theorem 2 keeps knitting affordable for at most logarithmically many subcircuits joined by single gates, and it is open whether any hard problem can be divided this way. Splitting into more subcircuits gives every balanced grouping of them a long boundary, and the theorem makes this exponentially expensive for every exact √ protocol only if the seam entropy of the target grows as n across that boundary. Excluding a family therefore needs such lower bounds, since a heuristic search for partitions can only show that a cheap cut exists, never that none does. The third asks how to prepare good states once the patches outgrow classical simulation. A single bridge keeps knitting affordable for any angles. A vanishing bridge angle, however, also decouples the patches, so affordability alone says nothing about the quality of the prepared state. Testing trainability and hardness together therefore needs scalable preparation and validation beyond classical references. Candidate computations for distributed quantum advantage should therefore be assessed jointly by their reconstruction cost, the classical difficulty of the required branch tasks, and, for variational approaches, gradient resolution, since none of these three properties can be inferred from the others.

VI.

ACKNOWLEDGMENTS

SD acknowledges funding from the ETH Zurich Quantum Center. MGG acknowledges funding from the EPSRC UK Quantum Technologies Programme under grant EP/T001062/1 and VeriQloud. LP is supported by the National Research Foundation, Singapore through the National Quantum Office, hosted in A*STAR, under its Centre for Quantum Technologies Funding Initiative (S24Q2d0009). LP is also partially supported by A*STAR under its Young Investigator Research Grant (YIRG) M25N8c0131. Claude Opus 5.5, Fable 5.1 (Anthropic) and GPT Astra (OpenAI) were used to draft and revise text, suggest references, write and debug analysis and plotting code, and check the manuscript for internal consistency. The tools also proposed two parts of the additional data analysis in the appendices, namely the transfer of angles between patch sizes (App. G), and the post-correction method used

15 Class

Signature

Meaning

Rows

Cuttable, no advantage

C1 ✓, S1 ✗

A cheap seam with easy subcircuits. By Theorem 1 the knitted computation is classically easy, so the verdict is structural rather than a suspicion. We expect most bounded-interface proposals to fall here.

7, 8a, 11b

Advantage, not cuttable

S1 ✓, C2 ✗

Seam entropy grows with the boundary, so Γcut = 2Ω( n) in two dimensions. By Proposition 1 no exact knitting protocol does better, so the verdict does not reflect a poor decomposition.

11a

Optimally cut, unaffordable

C2 ✗, Γcut = Γsep

The cost is known exactly but exceeds the budget. The trans- 8b, 12a parent toric-code seam costs 2L − 1, the same as the optimal joint cut of L − 1 wires with classical communication (App. A). These families calibrate the numerics.

Cheaply cuttable, still hard

C1, T1, S1 ✓

A bounded interface, bounded local depth, and hardness inside a subcircuit hold at once. The three conditions are independent (Sec. V). Corollary 7 shows that the class is not empty.

√

11c, and 12b if S1 holds

TABLE III. The four classes of screening outcomes. The signature lists the criteria of Table I that determine each class, and the last column gives examples from Table II. The target of a distributed FLDC is the last class.

at h = 0 (App. J 4). The authors reviewed all outputs and take full responsibility for the content. The research questions, results, proofs, and experimental design are the authors’.

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DATA AND CODE AVAILABILITY

The code used to generate the circuits and analyze the data will be made publicly available upon publication and is available from the corresponding author upon request.

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CONTENTS

A. Knitting cost and seam entanglement 1. The cost–entropy identity 2. Bounded interfaces 3. Transparent toric-code seam 4. Scaling regimes and affordable interface sizes

19 19 20 21 23

B. Classical simulability 1. Knitting easy subcircuits is easy 2. Sufficient conditions for easiness

23 23 24

C. Trainability 1. The gradient bound and its hypotheses 2. Barren plateaus from unitary designs 3. Gradient estimation through a cut

25 25 25 25

D. Hardness at a bounded interface

26

E. Tier-1 screen of explicit generators

27

F. Model, circuit, and exact endpoints 1. Encoder and sweeps 2. Zero field 3. Zero-field angles at nonzero field 4. Polarized limit

28 29 29 29 30

G. Transferring angles between patch sizes 1. Why angles might transfer between sizes

30 30

19 2. Transfer rules 3. Performance of the transfer rules

32 34

H. Knitted estimators and seam entropy 1. Energy and contact correlation 2. Bridge angle 3. Seam entropy

36 36 36 38

I. Gradient-variance diagnostic

38

J. Hardware circuits and settings 1. Circuit coordinates and trained angles 2. Compilation and error suppression 3. Signal loss and its depth dependence 4. Correction calibrated at zero field

40 40 41 41 42

Appendix A: Knitting cost and seam entanglement

This appendix proves Proposition 1 and Theorem 2 and collects every bound on Γcut used by criteria C1 and C2. The cost–entropy identity rests on the robustness of entanglement of pure states [48], and the scaling regimes combine it with the norms of Eqs. (4) and (5). Throughout, ρA = TrB ρ, logarithms are base two, and S(ρ) = − Tr ρ log2 ρ. 1.

The cost–entropy identity

We define the cost of a cut at the level of the output state and relate it to the circuit-level norm Γcut and to the seam entropy of Eq. (6). Definition 2 (State-level cutting cost). For a bipartite state ρAB , nP o P A B A B Γsep (ρ) = inf |c | : ρ = c σ ⊗ τ , σ , τ states . i i i i i i i i

(A1)

Lemma 1 (Hierarchy). Let a circuit with product input |0n ⟩ have its cross-device channels replaced by quasiprobability decompositions of the form of Eq. (1) whose terms are local operations with classical communication, possibly including measurements whose outcomes multiply the estimator by a sign, and let Γcut be the infimum of their coefficient norms. Then Y Γsep (ρout ) ≤ Γcut ≤ γa , (A2) a

where ρout is the output of the uncut circuit and γa are the norms of independent per-channel decompositions. P Proof. Substituting a decomposition with coefficients ai into the circuit gives ρout = i ai (ρi,+ − ρi,− ), where ρi,± is the unnormalized state prepared by branch i when the product of its outcome signs is ±1. Each branch is an instrument whose outcomes sum to a trace-preserving map, so Tr ρi,+ + Tr ρi,− = 1. It acts on a product input by subcircuit-local circuits interleaved with local operations and classical communication, so each ρi,± is separable, that is, a nonnegative combination of product states with weights summing to Tr ρi,± . Inserting these expansions gives P a decomposition of the form of Eq. (A1) with coefficient norm i |ai |. Taking infima gives the first inequality; the product of independent per-channel decompositions is one admissible decomposition, which gives the second. Theorem 3 (Exact pure-state cutting cost). For any state ρ, Γsep (ρ) = 1 + 2R(ρ), where R is the robustness of entanglement relative to separable states [48]. For a pure state ψ with Schmidt probabilities {λa }, X p 2 Γsep (ψ) = 2 λa − 1 = 2 · 2S1/2 (ρA ) − 1. (A3) a

Proof. Let ρ = i ci σiA ⊗τiB . Taking traces, i ci = 1. Collecting positive and negative coefficients, ρ = p+ ω+ −p− ω− with separable states ω± , p+ − p− = 1, and coefficient norm p+ + p− = 1 + 2p− . Conversely, any such two-term form is a valid decomposition once ω± are expanded into products. Hence Γsep = 1 + 2 min p− , and min p− is by definition the robustness R(ρ): the least weight of a P separable state ω− such that (ρ + p− ω− )/(1 + p− ) is separable. For pure √ states Vidal and Tarrach showed R(ψ) = ( a λa )2 − 1 [48], which gives Eq. (A3). P

P

20 Proof of Proposition 1. The identity is Theorem 3, and the inequality is Lemma 1. For pure targets the identity reads S1/2 (ρA ) = log2 [(Γsep + 1)/2], so state-level cutting cost and seam entropy determine each other. Attaining Γsep may require a decomposition that is hard to find, but the lower bounds below do not depend on finding it. Corollary 1 (Area-law no-go). Let {ρn } be pure outputs of distributed circuits. If Γsep (ρn ) = poly(n), in particular if Γcut = poly(n), then S(ρA ) ≤ S1/2 (ρA ) = log2

Γsep + 1 = O(log n). 2

(A4)

Hence no family whose targets carry seam entanglement S(ρA ) = Θ(L) with L = ω(log n) is efficiently cuttable. Conversely, S1/2 = O(log n) implies Γsep = poly(n). Proof. Combine Lemma 1 and Theorem 3 with S ≤ S1/2 , which holds because Rényi entropies do not increase with their index. Subcircuits are executed unmodified, so the entanglement structure inside each subcircuit, including any area law within it, is untouched by cutting; the corollary concerns only the entanglement that reconstruction must supply across the seam. Read in the other direction, a cut of κ channels with per-channel norm at most γ̄ gives S1/2 (ρA ) ≤ log2 [(γ̄ κ + 1)/2] ≤ κ log2 γ̄, so a bounded interface forces bounded seam entropy. Subcircuits run unmodified, so the entanglement inside each of them is untouched by cutting. The corollary concerns only the entanglement that knitting must supply across the seam. Read in the other direction, a cut of κ gates with per-gate norm at most γ̄ gives S1/2 (ρA ) ≤ log2 [(γ̄ κ + 1)/2] ≤ κ log2 γ̄, so a bounded interface forces bounded seam entropy. Example 1 (Long-range entanglement is p not the p obstruction). The n-qubit GHZ state split into two halves has Schmidt probabilities { 21 , 12 }, hence Γsep = 2( 1/2 + 1/2)2 − 1 = 3, independently of n. More generally, any state of Schmidt rank R across the seam has Γsep ≤ 2R − 1. Long-range entanglement alone does not determine the cost; seam capacity does. A cheap cut constrains the whole Schmidt spectrum across the seam, not only its entropy. √ Corollary 2 (Cheap seams are low-rank seams). Let ψAB be pure with Schmidt coefficients { λa } across A:B, and let ε ∈ (0, 1]. There is a set K of Schmidt indices with |K| ≤

2S1/2 (ρA ) Γsep (ψ) + 1 Γcut + 1 = ≤ ε2 2ε2 2ε2

(A5)

such that the normalized truncation ψK of ψ to K satisfies ∥ψ − ψK ∥1 ≤ 2ε. Hence |⟨O⟩ψ − ⟨O⟩ψK | ≤ 2ε∥O∥ for every observable, and an efficiently cuttable family has a seam of polynomial effective Schmidt rank at any fixed accuracy. P √ √ Proof. Write F = a λa = 2S1/2 /2 ≥ 1 and keep K = {a : λa ≥ η} with η = ε2 /F ≤ 1. Every kept index has √ √ P √ √ √ P √ 2 2 λa ≥ η, and the λa sum to F , so |K| ≤ F/ η = F /ε . The discarded weight obeys a∈K λa ≤ η a λa = ε2 , / √ √ so the retained weight is p ≥ 1 − ε2 and |⟨ψ|ψK ⟩| = p. For pure states ∥ψ − ψK ∥1 = 2 1 − p ≤ 2ε. The remaining forms of Eq. (A5) follow from Eq. (10) and Γsep ≤ Γcut . The corollary concerns the state, not an algorithm. Estimating ⟨OA ⊗ OB ⟩ from the truncated state would require transition elements between Schmidt vectors, which the branch tasks do not provide. The stitching theorem below avoids the Schmidt basis by reusing the cutting decomposition, whose branches are already products of subcircuit states.

2.

Bounded interfaces

The single bridge of Sec. IV is the case κ = 1 of the following two results, which bound the seam entropy and the knitting cost of any circuit whose subcircuits interact through κ Pauli rotations.

21 Proposition 2 (Capacity bounds seam entropy). Let U consist of subcircuit-local gates and κ contact gates exp[−iϕj PjA PjB /2], with PjA and PjB Pauli strings supported in A and B, applied to a product input. The state of Eq. (15) has this form, with κ = 1. The output then has Schmidt rank at most 2κ across A:B, and S1/2 (ρA ) ≤ κ,

Γsep (ψ) ≤ 2 · 2κ − 1,

(A6)

uniformly in the angles ϕj , in n, and in the subcircuit-local gates. Proof. Each contact gate equals cos(ϕj /2) 11 − i sin(ϕj /2) PjA ⊗ PjB , an operator of Schmidt rank at most two across A:B, so it at most doubles the Schmidt rank of the state. Subcircuit-local gates leave the rank unchanged, and√the PR √ product input has rank one, so the output rank is R ≤ 2κ . The Cauchy–Schwarz inequality gives a=1 λa ≤ R, hence S1/2 ≤ log2 R ≤ κ, and Eq. (10) gives the bound on Γsep . The bound follows from the operator Schmidt rank of the contact gates alone, so it holds before any decomposition is chosen. Corollary 1 reaches a similar bound from the opposite direction, through the cutting cost. Proposition 3 (Cost of cutting bounded interfaces). The contact gate exp[−iϕ P A P B /2] admits a local quasiprobability decomposition with γ(ϕ) = 1 + 2| sin ϕ|, optimal among local decompositions with or without classical communication [14, 19]. Cutting κ contacts independently therefore costs Γcut (ϕ) ≤

κ Y

P  (1 + 2| sin ϕj |) ≤ min 3κ , e2 j |ϕj | ,

(A7)

j=1

and near a warm start, ϕj → 0, the overhead tends to one: weakly coupled contacts are asymptotically free, consistently with the seam entropy they generate also vanishing with ϕ.

3.

Transparent toric-code seam

We now turn to the opposite geometry, a single toric-code patch divided by a straight cut across which anyons can pass anywhere along the seam. This is the seam of rows 8b and 12a of Table II. Two lemmas turn a witness operator into a lower bound on the coefficient norm. Applied to the stabilizers straddling the seam, they give an exponential lower bound certified by operators of bounded support, together with the exact cost of the seam. Lemma 2 (Witness ratio). For any Hermitian W , Γsep (ρ) ≥

| Tr(W ρ)| . supσA ,τ B | Tr[W (σ A ⊗ τ B )]|

(A8)

The same bound holds for the coefficient norm of any signed product decomposition of any operator whose W -expectation matches Tr(W ρ) to additive error ε, with numerator | Tr(W ρ)| − ε. P P Proof. | Tr(W ρ)| = | i ci Tr[W (σi ⊗ τi )]| ≤ i |ci | sup | Tr[W (σ ⊗ τ )]|. Lemma 3 (Maximally entangled overlap). Let ΦD be maximally entangled of Schmidt rank D across A:B. For any state ω of Schmidt number r, in particular any product state with r = 1, ⟨ΦD |ω|ΦD ⟩ ≤ r/D. p P P Proof. For a pure state |φ⟩ = b≤r sb |ub ⟩|vb ⟩, each |⟨ΦD |ub vb ⟩| ≤ D−1/2 , so |⟨ΦD |φ⟩| ≤ D−1/2 b sb ≤ r/D by the Cauchy–Schwarz inequality. Mixed states of Schmidt number r are mixtures of such states. Consider a planar toric-code patch of Wp × Hp plaquettes with qubits on edges, a star operator on every vertex, truncated at the boundary, and a plaquette operator on every face, so that its stabilizers fix a unique state ψ. Definition 3 (Straddling stabilizer group). For a vertical seam through vertex column c, 0 < c < Wp , assign to A every qubit whose edge midpoint has x < c and to B the rest, and let L = Hp + 1 be the number of seam vertices. Let G∂ be the set of star and plaquette operators whose support meets both A and B, and define the strip witness W =

Y 11 + g g∈G∂

2

.

Theorem 4 (Strip-local robust seam witness). For the state ψ of Definition 3:

(A9)

22 (i) (locality) |G∂ | = 2L − 1, consisting of the L stars centered on the seam and the L − 1 plaquettes immediately to its left. Every element acts on at most four qubits within one lattice spacing of the seam and is localized at a single seam position, and all elements commute, so W is a product of 2L − 1 commuting projectors of bounded support; (ii) (normalization) Tr(W ψ) = 1; (iii) (separable ceiling) supσA ,τ B Tr[W (σ A ⊗ τ B )] ≤ 2−(L−1) ; (iv) (robust bound) any signed product decomposition, in particular any cutting protocol of Lemma 1, whose reconstruction reproduces 2L−1 . P⟨W ⟩ to additive error ε obeys Γ ≥ (1 − ε) −(2L−1) 2L−1 Moreover W = 2 products of seam-crossing star and plaquette g∈⟨G∂ ⟩ g is the uniform average of the 2 operators, the closed e- and m-string configurations threading the seam, so reproducing all of those correlators to error ε reproduces ⟨W ⟩ to error ε. Proof. (i) The stars centered at the L seam vertices have one edge in A and the rest in B; the plaquettes in the column immediately left of the seam have three edges in A and one in B; every other star or plaquette lies wholly on one side. Stars and plaquettes share an even number of edges, so G∂ is abelian. (ii) Each g stabilizes ψ. (iii) Define the alternating form ω(g, h) on the F2 space ⟨G∂ ⟩ recording whether the restrictions g|A and h|A anticommute. The star at seam row y restricts to X on the single horizontal edge ey entering the seam at that row, and the plaquette at row y ′ restricts to Z on ey′ , ey′ +1 and one vertical edge. Hence ω(stary , plaqy′ ) = δy,y′ + δy,y′ +1 , a bidiagonal L × (L − 1) matrix of rank L − 1, while stars and plaquettes restrict to X-type and Z-type operators that commute among themselves. So ω has rank 2(L − 1). Choose a symplectic basis (ui , vi )L−1 i=1 for ω and a complement spanning its radical, so that W = Π0

L−1 Y (11 + ui )(11 + vi ) (11 + ui )(11 + vi ) ≤ , 4 4 i=1 i=1

L−1 Y

(A10)

with Π0 the commuting projector associated with the radical. The A-restrictions of {ui , vi } obey the commutation relations of Pauli X and Z on L − 1 qubits, and because ui and vi commute, so do the B-restrictions. A local Clifford unitary VA ⊗ VB therefore maps the right-hand side to Φ2L−1 ⊗ 11 with Φ2L−1 a maximally entangled projector [91]. Local unitaries map products to products, and partial traces of products are products, so Lemma 3 gives the ceiling 2−(L−1) . (iv) Lemma 2. The symplectic basis used in (iii) is in general supported along the whole seam, but it enters only the proof of the ceiling. By (i) the witness itself is a product of commuting projectors of bounded support, each attached to one seam site; a disentangling circuit never has to be constructed, only shown to exist. Corollary 3 (Exact transparent-seam cost). For the state of Definition 3, ρA has a flat spectrum of rank 2L−1 , so Γsep (ψ) = 2 · 2L−1 − 1 = 2L − 1, and every cutting protocol satisfies Γcut ≥ 2L − 1, with a sufficient shot count scaling as Γ2cut ∼ 4L . More generally, k Bell pairs have Γsep = 2k+1 − 1, which coincides with the coefficient norm of the optimal joint cut of k wires with classical communication [20], so that protocol is optimal already at the state level. Proof. For a stabilizer state, ρA is proportional to a projector and S(ρA ) is half the F2 rank of the anticommutation form of the A-restrictions of a full set of stabilizer generators [91]. A generator supported wholly in A or in B has a trivial form with every other generator, so only G∂ contributes, and step (iii) of Theorem 4 gives S(ρ PA )√= L − 1 with a flat spectrum. Then S1/2 = L − 1, and Theorem 3 and Lemma 1 give the claims. For k Bell pairs, ( a λa )2 = 2k . Corollary 4 (Seam-crossing loop degradation). Any signed product decomposition of coefficient norm Γ assigns the witness a value ⟨W ⟩rec ≤ Γ 2−(L−1) . At fixed subexponential budget, the joint correlator of the seam-crossing string algebra therefore cannot be reproduced beyond a value exponentially small in L, whereas observables supported in one device are reproduced without overhead, since subcircuits run unmodified. Because W is a witness rather than a fidelity, it applies away from the fixed point. The overlap |⟨ψ0 |ψh ⟩|2 between the fixed-point and field-perturbed states decays exponentially in the total system size, whereas Tr(W ρh ) is controlled by a strip of width O(1) around the seam. Corollary 5 (Perturbed reduction). For any state ρ on the patch, a decomposition reproducing ⟨W ⟩ρ has Γ ≥ Tr(W ρ) 2L−1 . Hence if the field-perturbed ground state obeys Tr(W ρh ) ≥ e−c(h)L , then Γ ≥ 2(1−c(h)/ ln 2)L−1 , and the exponential bound survives at every field with c(h) < ln 2. Proof. Lemma 2 with numerator Tr(W ρ) and Theorem 4(iii).

23 4.

Scaling regimes and affordable interface sizes

This subsection proves Theorem 2. Let {Un } be cut into D(n) subcircuits of width w(n), severing K(n) gates in total. Two lower bounds on Γcut apply, one for rotation gates and one for every exact protocol. For the first, let the severed gates be two-qubit rotations with | sin θs | ≥ c > 0 on a fraction f > 0 of them. QK Independent cutting gives Γcut = s=1 (1 + 2| sin θs |) ≥ (1 + 2c)f K = 2Ω(K) by Eq. (4). When the time-slice condition holds, the joint decomposition of Eq. (5) gives Γcut ≥ 2(1 + c)f K − 1 = 2Ω(K) . A connected arrangement of D subcircuits has K ≥ D − 1, so both costs are 2Ω(D) . For the second, group the subcircuits into two halves A and B and write s(n) = S1/2 (ρA ) for the pure target. Proposition 1 gives Γcut ≥ 2 · 2s(n) − 1 for every exact protocol. Proof of Theorem 2. If D = O(1) and w = Θ(n), the total interface K may stay bounded, and then Γcut ≤ 3K = O(1) by Eq. (4). The asymptotics in n then describe subcircuits that grow behind a fixed interface. If w = O(1) and Ω(D) D = Θ(n/w), then K ≥ D − 1 = Ω(n/w), and the first bound gives 2√ under its angle condition. Independently, a √ target with s(n) = Ω( n) across a balanced grouping has Γcut = 2Ω( n) by the second bound. The entropy hypothesis holds, for example, for a geometrically local two-dimensional target whose √ entanglement grows with the boundary. Every balanced grouping of the subcircuits then has a boundary ∂ of Ω( n) qubits, and √ S(ρA ) ≥ a|∂| − b with a > 0 gives s(n) ≥ S(ρA ) = Ω( n), since S1/2 ≥ S. An area-law upper bound alone does not imply the hypothesis, which is why Theorem 2 states it explicitly. We now evaluate√Eq. (9) for representative budgets. We take ε = 0.01 and δ = 0.05, for which Eq. (8) gives Γmax ≈ 3.68 × 10−3 M . Table IV lists the resulting counts. Shot budget M

106

109

1012

Γmax κjoint max , one time slice κind max , independent gates

3.7 1 1

116 5 4

3682 10 7

TABLE IV. Largest affordable number of severed two-qubit rotations from Eq. (9), for additive error ε = 0.01 and failure probability δ = 0.05, using the worst-case norm of each construction as a sufficient norm-budget threshold .

Each additional severed gate costs a factor of nine in shots for independent decompositions and a factor approaching four for joint ones. The affordable count therefore grows only as 12 log2 M and remains a small integer for any realistic budget. Interface gates interleaved through the circuit, rather than confined to one time slice, must use the independent count. For the single bridge gate of Sec. IV, κ = 1 and the joint and independent norms coincide. Appendix B: Classical simulability

This appendix proves Theorem 1, which underlies criterion S1, and gives the two sufficient conditions for efficient simulation that criteria S2 and S1 use. The theorem formalizes an observation of Refs. [11, 19], that knitting classically simulable subcircuits yields a classically simulable estimate, with the branch tasks as the unit of hardness. 1.

Knitting easy subcircuits is easy

This subsection proves Theorem 1. Each branch of a cutting decomposition without classical communication is a product of subcircuit states, so a global expectation value is a signed sum of products of subcircuit expectation values. Theorem 5 (Stitching). Let Un carry a cut of κ(n) gates, each replaced by at most m = O(1) local terms of total coefficient norm at most γ̄, none requiring classical communication. Then X X |cα | = Γ ≤ γ̄ κ , #{α} ≤ mκ , (B1) ρout = cα σαA ⊗ ταB , α

α

where σαA and ταB are prepared by the two subcircuits with the operations of branch α inserted and weighted by the signs of any measurement outcomes. Let O have support at most r = O(1) and ∥O∥ ≤ 1. Suppose the branch tasks,

24 estimating Tr(O′ σαA ) and Tr(O′′ ταB ) to additive error η for any α and any O′ , O′′ of support at most r and norm at most one, are solvable classically in time T (n, 1/η). Then ⟨O⟩ can be estimated to additive error ε in time
O 4r mκ T (n, 3 · 4r Γ/ε) .

(B2)

In particular, if κ = O(log n) and both branch tasks are in BPP, then estimating ⟨O⟩ to inverse-polynomial additive error is in BPP. P Proof. Expand O = j oj PAj ⊗ PBj in the Pauli basis of its support, with at most 4r terms and |oj | ≤ ∥O∥ ≤ 1. By P P Eq. (B1), ⟨O⟩ = j oj α cα Tr(PAj σαA ) Tr(PBj ταB ). Estimate each of the 2 · 4r mκ factors to additive error η ≤ 1. The exact factors lie in [−1, 1], so each product carries error at most 2η + η 2 ≤ 3η, and the total error is at most 3 · 4r Γη. Choosing η = ε/(3 · 4r Γ) gives error ε, and the running time is the number of factors times the cost of each. For randomized subroutines, repeating each call O(log(4r mκ /δ)) times and taking the median makes all estimates succeed together with probability at least 1 − δ. If κ = O(log n), then mκ and Γ ≤ γ̄ κ are polynomial, so the branch tasks are called polynomially often at inverse-polynomial precision. Proof of Theorem 1. This is the last statement of Theorem 5. Corollary 6 (The trench coat). Under the hypotheses of Theorem 5 with κ = O(log n), if estimating local expectation values of {Un } is not in BPP, then the branch task of at least one subcircuit is not in BPP. A family that is efficiently cuttable by such a protocol, and whose subcircuits are both classically easy, is therefore classically easy as a whole. The theorem uses only the branch tasks of criterion S1, with no Schmidt basis and no transition amplitudes. It covers structured protocols with boundedly many local terms per severed gate and no classical communication, which includes the six-branch decomposition of the bridge gate used in Sec. IV. Allowing classical communication with a message alphabet of constant size per gate multiplies the branch count by a further factor O(1)κ and leaves the conclusion unchanged. A protocol reaching Γ = poly(n) with superpolynomially many branches is not covered. Corollary 2 still shows that its seam has polynomial effective Schmidt rank, but turning that into a simulation would require transition elements, so this case remains open.

2.

Sufficient conditions for easiness

Proposition 4 (Light-cone easiness). Let U be a geometrically local circuit on a two-dimensional lattice with conventional depth d, and let O be an observable supported in a region of diameter r with ∥O∥ ≤ 1. Then ⟨0n |U † OU |0n ⟩ 2 can be computed in time 2O((r+d) ) , independent of n. In particular, constant-depth two-dimensional families fail S2. Proof. The backward light cone of supp(O) under d layers of geometrically local gates is contained in a disk of radius O(r + d), hence contains O((r + d)2 ) qubits. Gates outside the cone cancel between U and U † , and a state-vector simulation of the cone circuit on its product input gives the result. The argument uses only the number c of qubits in the backward light cone of supp(O) in the gate DAG, and gives cost poly(n) 2O(c) for any circuit; this is the quantity S2 measures. A sequential FLDC can have conventional depth Θ(n) and cones that grow with n, so bounded local depth does not imply a bounded cone. Proposition 5 (Quasi-one-dimensional easiness). Let a circuit of two-qubit gates act on a w × n strip, with gates crossing any vertical cut at most t times in total. Then the output has Schmidt rank at most 4t across every vertical cut, admits a matrix-product description of bond dimension 2O(w+t) , and its local expectation values can be computed classically in time poly(n) 2O(w+t) . For depth d, t ≤ wd, so wd = O(log n) implies polynomial cost. Constant-p QAOA on graphs of bounded maximum degree ∆ is likewise easy for local observables, since the backward causal cone of an r-local observable contains at most r∆O(p) qubits [59]. Proof. A two-qubit gate crossing a cut has operator Schmidt rank at most four across it, so it multiplies the Schmidt rank of the state by at most four, and gates not crossing the cut leave it unchanged; this gives 4t at every time. Ordering the qubits column by column, a cut inside a column exceeds the adjacent vertical cut by at most w qubits, which gives the bond dimension, and a time-evolving matrix-product simulation [92] gives the running time. The QAOA statement is the light-cone argument of Proposition 4 applied on the interaction graph.

25 Appendix C: Trainability

This appendix states the finite-local-depth gradient bound behind criteria T1 and T2 with its hypotheses, gives the sufficient condition for a barren plateau behind the fail conditions of T2 and T3, and shows that gradients can be estimated through a cut without losing the bound. 1.

The gradient bound and its hypotheses

Recall from Definition 1 that U (θ) = UM (θM ) · · · U1 (θ1 ) consists of blocks that act on at most β qubits each, with at most χ blocks acting on any one qubit. For X C(θ) = λν ⟨Oν ⟩θ , | supp(Oν )| ≤ r = O(1), ν

and a parameter in block Uk(µ) , the bound used in Sec. II is Varθ [∂µ C] ≥ g, g = 2 4−rχβ

X

λ2ν .

(C1)

ν: supp(Oν ) ∩ supp(Uk(µ) )̸=∅

The bound holds under the hypotheses of Zhang, Liu, and Zhang [41]: the blocks are drawn independently from local 2-designs, θµ parametrizes a gate inside block Uk(µ) , and only cost terms whose support overlaps that block contribute to the sum. The bound g depends on the parameter µ even though that index is suppressed in the notation. The quantities β and χ in Eq. (C1) need some care. They are not the circuit depth. A sequential FLDC can have conventional depth Θ(n) while β and χ stay bounded, as for the toric-code circuits of Table II, and this gap is what the FLDC construction exploits. They are also not intrinsic to a circuit, which has no canonical block decomposition. The finest choice, one block per gate, gives β = 2 with χ equal to the largest number of gates on any qubit, and coarser blocks trade a larger β for a smaller χ. Criterion T1 therefore records both quantities for a stated decomposition. Finally, every circuit has some β and χ, and an FLDC is distinguished only by both remaining bounded as n grows. 2.

Barren plateaus from unitary designs

Proposition 6 (Design-induced barren plateau). Let C(θ) = Tr[OU ρ0 U † ] with ∥O∥ ≤ 1, and write U = V2 e−iθG/2 V1 with G a Pauli string. If V1 or V2 is drawn from a unitary 2-design on the full n-qubit register, independently of the other, then E[∂θ C] = 0,

Var[∂θ C] ≤

Tr(O2 ) 2n ≤ n = 2−Ω(n) . n 4 −1 4 −1

(C2)

Consequently, FLDC families whose blocks are enlarged until the circuit, or the sub-circuit on either side of a parameter, forms a global 2-design lose trainability, and the guarantee of Eq. (C1), which carries the factor 4−rχβ , becomes vacuous as soon as rχβ = ω(log n). Proof. Write d = 2n , R = e−iθG/2 , O′ = V2† OV2 and σ ′ = RV1 ρ0 V1† R† . Differentiating, ∂θ C = Tr(Aσ ′ ) = Tr(O′ B) with the traceless Hermitian operators A = 2i [G, O′ ] and B = 2i [σ ′ , G], which obey Tr(A2 ) ≤ Tr(O2 ) and Tr(B 2 ) ≤ Tr(σ ′2 ) ≤ 1. For V drawn from a 2-design and traceless Y , the standard second-moment formulas [26] give E[Tr(V XV † Y )] = 0 and E[Tr(V XV † Y )2 ] = Tr(Y 2 )[Tr(X 2 ) − Tr(X)2 /d]/(d2 − 1) ≤ Tr(X 2 ) Tr(Y 2 )/(d2 − 1). If V1 is the design, apply this with V = RV1 , X = ρ0 and Y = A; if V2 is, apply it with V = V2† , X = O and Y = B. Both cases give Eq. (C2). The vacuous case is the fail condition of T2, and the straight line on a log plot is the fail condition of T3. 3.

Gradient estimation through a cut

The following result shows that cutting leaves the landscape and its gradient bound unchanged, and that the sampling cost of a gradient component is set by the interface norm. It is the guarantee that the diagnostic of App. I would test.

26 Theorem 6 (Cutting guarantee). Let U be a distributed FLDC with block support β and local depth χ, cut through κ channels with per-channel quasiprobability norms γa ≤ γ̄, and let O be an observable with ∥O∥ ≤ 1. The product Q estimator ob = Γ sgn(aα ) x, with Γ = a γa ≤ γ̄ κ , is unbiased with |b o| ≤ Γ, so Eq. (3) bounds the number of shots that resolve ⟨O⟩ to error ε. A gradient component given by the parameter-shift rule, ∂µ C = 12 [C(θµ + π2 ) − C(θµ − π2 )], is estimated to error ε by estimating both shifted costs to error ε. Removing interface gates deletes blocks, so neither subcircuit’s χ nor β increases, and cutting leaves the landscape pointwise unchanged, P so for blocks drawn from local 2-designs the boundPof Eq. (C1) [41] applies verbatim. Consequently, for a cost C = ν λν Oν with | supp(Oν )| ≤ r, ∥Oν ∥ ≤ 1 and Λ = ν |λν |, resolving ∂µ C to a constant fraction of its root-mean-square value is achieved with  X −1 1 N = O Λ2 γ̄ 2κ 4rχβ λ2ν log δ ν

(C3)

shots per shifted cost, with the sum as in Eq. (C1); this is independent of n whenever Λ is. Proof. Unbiasedness and |b o| ≤ Γ follow from sampling α with probability |aα |/Γ and |x| ≤ ∥O∥. The landscape statement is linearity of the reconstruction in expectation. For Pthe cost, each shot lies in [−ΛΓ, ΛΓ], and combining Eq. (3) at ε = Θ(v 1/2 ) with the variance bound v ≥ 2 · 4−rχβ ν λ2ν of Eq. (C1) gives the shot count. Appendix D: Hardness at a bounded interface

This appendix shows that the three structural properties targeted in the main text are mutually compatible. In particular, finite local depth and a bounded interface do not by themselves force the branch tasks to be classically easy. The result is an existence statement: it places a worst-case hard FLDC inside one subcircuit and connects it to another subcircuit through a constant-size interface. It does not identify a natural average-case hard instance. Theorem 7 (Subcircuit hardness at bounded interface). Local-observable estimation to constant additive accuracy for two-dimensional sequential FLDCs is BQP-complete in the worst case [77, Sec. V]. Consequently, for every constant κ there exists a κ-interface distributed FLDC family whose subcircuit-local estimation task remains BQP-hard after the interface gates are removed. Proof. Membership in BQP is immediate, since a quantum computer can run the corresponding subcircuit. (x) For hardness, let {UA } be the worst-case BQP-hard family of two-dimensional sequential FLDCs from Ref. [77]. (x) Each UA has bounded block support and bounded local depth. Place this family on subcircuit A. Let UB be any fixed FLDC on subcircuit B, and add a fixed set Uport of κ = O(1) contact gates between designated boundary ancillas of A and B. Define (x)

U (x) = (UA ⊗ UB ) Uport .

(D1)

The resulting distributed circuit is still an FLDC: the contact gates increase the local depth only on O(κ) boundary ancillas. The interface has constant size by construction. (x) Now remove the interface gates and consider the branch task on subcircuit A. It contains the original family {UA }. Therefore, a classical polynomial-time algorithm for the branch task would give a classical polynomial-time algorithm for the BQP-hard local observable problem of Ref. [77]. Hence the branch task is BQP-hard in the worst case. Corollary 7 (Coexistence). There exist two-dimensional distributed FLDC families with κ, χ, β = O(1) for which all of the following hold. (i) The interface is cheaply cuttable: Γcut ≤ 3κ = O(1), so O(9κ ε−2 log δ −1 ) shots suffice per bounded-norm observable. (ii) Under the local-2-design initialization assumptions of Ref. [41], local costs with n-independent coefficients have gradients with variance bounded below by X 2 4−rχβ λ2ν = Ω(1) ν

whenever the sum is bounded below.

27

(b) T2: guarantee constant (c)

T1: local structure

(d)

S2: light cone

C1: interface capacity

1 Cone fraction c=n

102

102

¡200

Formal log10 g

Local depth Â

0

¡400

101

¡600 8

32

n

128 512

3 Hardware-efficient ansatz 5 Routed QAOA (p = 2)

▿: g = 0

8

32

n

128 512

6 Brickwork: linear depth 7 Sequential MPS (D = 4)

Severed gates ∙found

(a)

0.5

0

∙max = 7 (M = 1012 )

101

100 8

32

n

128 512

8a Clifford toric: bounded contact 8b Clifford toric: extensive seam

8

32

n

128 512

12b Single-bridge FLDC Brickwork: constant depth (control)

FIG. 8. Tier-1 screen of eight explicit generators against the number of qubits n. (a) Local depth χ (T1). (b) Formal log10 g of Eq. (C1) (T2), with no curve for the Clifford generators, which have no variational blocks, and filled triangles at an arbitrary height where g = 0. (c) Fraction c/n of qubits in the largest probed backward causal cone (S2). (d) Severed gates κfound of the 12 partition found by the heuristic search (C1), an upper bound on κ∗ (n), with the independent-cut limit κind shots, max = 7 for 10 ε = 0.01, and δ = 0.05 (dashed) and the affordable region below it (shaded). All values are structural counts or the result of a seeded heuristic search, so there are no statistical error bars. All horizontal axes are logarithmic in base 2, the vertical axes of (a) and (d) are logarithmic, the panel titles name the criteria of Table I, and the legend below the panels applies to all four, with the row of Table II in front of each entry and (control) marking the constant-depth brickwork, which has no row.

(iii) The subcircuit-local observable-estimation task is BQP-hard in the worst case. (iv) Every state prepared by the architecture has seam entropy S1/2 ≤ κ, by Proposition 2. Thus bounded interface, finite local depth, and worst-case hard branch tasks can coexist. Conversely, a family whose target states have seam entropy growing with n cannot satisfy the first item for exact knitting, by Corollary 1. The trainability statement in item (ii) and the hardness statement in item (iii) refer to different kinds of evidence: random initialization for the former, and worst-case instances for the latter.

Appendix E: Tier-1 screen of explicit generators

This appendix illustrates the Tier-1 entries of Table II on explicit circuit generators, computing the quantities of criteria T1, T2, S2, and C1 and checking that they follow the trends recorded there. We screen eight generators with 7 to 517 qubits (Fig. 8 and Table V). Seven of them instantiate rows of Table II, and constant-depth brickwork serves as a shallow control. The single bridge is grown as a 1 × k strip, the bounded-contact toric code as two such strips joined by one contact gate, the extensive-seam toric code as patches from 2 × 1 to 22 × 11 plaquettes cut along a straight seam, and the chain generators and the hardware-efficient ansatz on chains and 3 × k grids. Routed QAOA applies p = 2 layers of all n(n − 1)/2 pairwise interactions through a linear SWAP network, as its recorded gate counts, exactly n(n − 1), show. Cones are backward gate-list light cones of up to twelve probed cost observables on the uncut generator, with every recorded gate, SWAP gates included, treated as acting on all its qubits, and they bound the cones of the cutting branches, whose inserted operations are local. Partitions come from a spectral bisection with Fiduccia–Mattheyses refinement, replaced by a multilevel variant at some sizes of the grid and seam generators, with 2 to 10 seeded restarts, and g is evaluated with the smallest sum of squared cost coefficients over parameterized blocks. All quantities follow directly from the generated gate lists, without state simulation, so they are exact properties of the generators, except for the balanced partition, which a seeded heuristic search provides. Local depth (T1). For the stated block decompositions, all generators have bounded block support, 2 ≤ β ≤ 4, so T1 reduces to checking whether χ remains bounded as n grows. The hardware-efficient ansatz (row 3) and linear-depth

28 TABLE V. Tier-1 screen of Fig. 8, with the trend recorded in Table II next to the value found at the largest generated size. All values are taken at about 512 qubits, 513 for row 3 and 517 for row 8b, except for routed QAOA, whose largest size is 192 qubits. The swatch gives the line of each generator in the figure. Boxed values fail the criterion or, for T2, collapse with the size. Row Generator

3 5 6a 7 8a 8b 12b –

Hardware-efficient ansatz Routed QAOA, p = 2 Brickwork, linear depth Sequential MPS, D = 4 Clifford toric, bounded contact Clifford toric, extensive seam Single-bridge FLDC Brickwork, constant depth

χ (T1)

log10 g (T2)

c/n (S2)

trend value

trend

value trend value

Θ(d) ω(1) ω(1) O(1) ≤6 ≤6 O(1) –

≤10−28

✓ ? ✓ ✓ ✓ ✓ ✓ –

342 384 256 3 3 2 18 4

−411 4−ω(1) g = 0 4−ω(1) −616 D−O(1) −10 – – – – – −86 – −9

κfound (C1) trend

value

1.00 ω(1) 1.00 Θ(n) √ 0.96 Θ( n) 1.00 O(1) 0.83 O(1) 1.00 Θ(L) 1.00 1 0.02 –

258 192 128 2 1 13 1 2

brickwork (row 6a) fail this check because their increasing layer counts give χ = Θ(n). Routed QAOA (row 5) also has χ = Θ(n), but here the growth comes from routing swaps, despite the fixed number of QAOA layers, p = 2 (Fig. 8(a)). Both Clifford toric-code generators (rows 8a and 8b) have bounded local depth, although their lack of variational parameters makes them structural comparisons rather than trainability tests. In contrast to the three growing-depth generators, the single-bridge FLDC (row 12b), sequential MPS preparation at fixed D = 4 (row 7), and constant-depth brickwork (control) have χ = O(1) and pass T1. Guarantee constant (T2). For the hardware-efficient ansatz (row 3) and linear-depth brickwork (row 6a), growing χ suppresses the formal bound through 4−rχβ , giving log10 g ≈ −411 and −616, respectively, at the largest sizes (Fig. 8(b)). Routed QAOA (row 5) instead gives g = 0 for parameters whose blocks overlap no cost term, making the coefficient sum in Eq. (C1) vanish. Neither a collapsing nor a zero lower bound establishes a barren plateau. The single-bridge FLDC (row 12b) has bounded local depth but does not satisfy the local-2-design assumptions: the encoder is fixed, and the bridge and anchor are not drawn from local 2-designs. Its formal value g ∼ 10−86 is therefore only formal, and trainability is assessed directly through T3 (Sec. IV C and App. I). The Clifford generators (rows 8a and 8b) have no variational parameters, so the Train criteria do not apply. In contrast, sequential MPS preparation at fixed D = 4 (row 7) and constant-depth brickwork (control) pass the scaling check: their formal bounds remain nonzero and independent of n, with log10 g ≈ −10 and −9, respectively. Under the local-2-design assumptions, these would provide size-independent variance lower bounds, although their small magnitudes do not ensure gradient resolution within a practical shot budget. Light cone (S2). Constant-depth brickwork (control) is the only family which fails the cone-growth check: its largest probed cone saturates at 10 qubits for n ≥ 12, permitting efficient classical simulation by Proposition 4 (Fig. 8(c)). All other generators have extensive probed cones, but this does not establish hardness, as the classically simulable Clifford codes (rows 8a and 8b) and fixed-bond MPS preparation (row 7) illustrate. Interface (C1). We compare the heuristic upper bound κfound ≥ κ∗ (n) with the independent-cut threshold κind max = 7 of Eq. (9), for M = 1012 shots, ε = 0.01, and δ = 0.05 (Fig. 8(d)). The growing-local-depth generators (rows 3, 5, and 6a) exceed this threshold by n = 32, and the extensive-seam Clifford code (row 8b) near n = 200. In contrast, the single-bridge FLDC (row 12b) and bounded-contact Clifford code (row 8a) pass with κfound = 1, while fixed-bond MPS preparation (row 7) and constant-depth brickwork (control) pass with κfound = 2 at every tested size. Exceeding this sufficient threshold does not establish unaffordability: angle-dependent norms, a joint decomposition where applicable, or a better partition may still satisfy C1. The Tier-1 screen illustrates the independence of local depth, interface size, and cone growth. Bounded local depth does not imply a bounded interface or cone, while extensive cones can coexist with classically easy branch tasks.

Appendix F: Model, circuit, and exact endpoints

The test of Sec. IV rests on exactly known states, and this appendix derives them. We first show that the circuit, built from the encoder and the pair blocks of the sweeps, has bounded local depth and is therefore an FLDC. At h = 0, we find the exact ground state and the angles that prepare it in closed form, which also provides the contact benchmark of Sec. IV D. Since we reuse the zero-field angles at nonzero field to avoid training, we also give their energy there in closed form and compare it with the ground energy. At h = 1, the ground state is again known in closed form, which gives an exact reference at the other end of the field sweep.

29 1.

Encoder and sweeps

A star and a plaquette share zero or two edges, so all stabilizers commute. A patch has (nx + 1)(ny + 1) stars and nx ny plaquettes, n + 1 generators with one relation, since the product of all stars is the identity. The stabilizer group therefore has rank n, and an isolated patch at zero field has a unique ground state |Ωp ⟩, the common +1 eigenstate of all stars and plaquettes. Since |0n ⟩ is a +1 eigenstate of every plaquette, applying the independent star projectors prepares it, Ep |0n ⟩ = |Ωp ⟩ =

Y 11 + Sv,p √ |0n ⟩, 2 v̸=v

(F1)

0

where v0 is the top-right vertex. The factor 2−1/2 per star normalizes the state, because distinct products of stars map |0n ⟩ to orthogonal basis states. A circuit for Eq. (F1) treats the vertices row by row from the bottom-left corner. For each v ̸= v0 it applies a Hadamard gate to a pivot spin on the edge from v to its right or upper neighbor, followed by controlled-NOT gates from the√pivot to the other spins of Sv,p . The pivot lies on no earlier star, so it is still in |0⟩, and the step applies (11 + Sv,p )/ 2. Every spin enters at most two such steps, so the encoder has bounded local depth. The encoder used in the simulations follows Ref. [87] and prepares the same state. For two consecutive edge spins (u, v), the pair block of Fig. 4(b) is uv v u Uuv (ϑ) = RZZ (ϑ5 ) RZ (ϑ4 ) RZ (ϑ3 ) RYv (ϑ2 ) RYu (ϑ1 ),

(F2)

with RP (ϑ) = exp(−iϑP/2) for a Pauli string P . The local depth of Sec. IV A counts the blocks acting on each spin in the generated circuits. It is dominated by the sixteen pair blocks that four sweeps apply to an interior spin, and it does not grow with the length of the patch, which we checked on strips of up to 512 spins. 2.

Zero field

Let |Ω0 ⟩ be the product of the two patch ground states, so that Hs |Ω0 ⟩ = (N + 2)|Ω0 ⟩. The contact Q anticommutes with the two stars incident to a and the two incident to b, and commutes with every other stabilizer. Hence |Ω1 ⟩ = Q|Ω0 ⟩ is orthogonal to |Ω0 ⟩, with Hs |Ω1 ⟩ = (N − 6)|Ω1 ⟩. Their span is invariant under H(0, t), which acts on it as   −4 −t H(0, t) = (−N + 2) 11 + . (F3) −t 4 √ With Dt = 16 + t2 and tan ϕ = t/4, the lower eigenpair is E0 (0, t) = −N + 2 − Dt ,

|Ω(0)⟩ = cos

ϕ ϕ |Ω0 ⟩ + sin |Ω1 ⟩. 2 2

(F4)

Relative to the bridge-off state |Ω0 ⟩, whose energy is −N − 2, the coupling lowers the energy by Dt − 4 ≈ 0.031 at t = 0.5, independently of the size. It is the anticommute with Q Pglobal ground state. The four stars S1 , . . . , S4 thatP form, with the contact, the part K = − i Si − tQ, and the anticommutation gives K 2 = ( i Si )2 + t2 11 ≤ Dt2 11, so K ≥ −Dt . Every other stabilizer contributes at least −1, and Eq. (F4) attains all these bounds, which proves optimality and uniqueness. ab The circuit prepares this state exactly. The bridge gives RZZ (ϕ)|Ω0 ⟩ = cos ϕ2 |Ω0 ⟩ − i sin ϕ2 |Ω1 ⟩, with the right amplitudes but the wrong relative phase. A star incident to a has eigenvalue +1 on |Ω0 ⟩ and −1 on |Ω1 ⟩, so the anchor rotation at α = π/2 corrects the phase and yields e−iπ/4 |Ω(0)⟩. Every single-spin Pauli operator has zero expectation value in |Ω(0)⟩, because it anticommutes with a stabilizer of each component and cannot map one component onto the other. In particular za = zb = 0, and since Q exchanges the two components, the contact correlation of Eq. (19) is Cab = ⟨Q⟩ = sin ϕ = t/Dt at every size. 3.

Zero-field angles at nonzero field

The zero-field angles prepare the same state at every field. Since ⟨Hf ⟩ = 0 and ⟨Hs ⟩ = N − 2 + 16/Dt in this state, their energy is exactly linear in h,  16  Ezf (h, t) = −N + 2 − Dt + h N − 2 + , (F5) Dt

30 which is the value the device should return for these angles. By the Hellmann–Feynman identity ∂h E0 = ⟨∂h H⟩ with ∂h H = Hs − Hf , this line is also the first-order expansion of the exact ground energy around h = 0. It is a variational upper bound for all h and differs from the ground energy at second order in h. Relative to the references of Sec. IV B, its excess at N = 48 is 9.1 × 10−3 , 0.051, 0.091, and 0.14 per spin at h = 0.1, 0.2, 0.25, and 0.3, respectively. The excess decreases slowly with N , a boundary effect; for example, at N = 80 it is 0.008 per spin at h = 0.1 and 0.081 per spin at h = 0.25. With no dressing and α = π/2, the knitting decomposition inserts only Pauli operators, Z measurements, and RZ (±π/2) rotations, so every branch circuit is a Clifford circuit at every field. 4.

Polarized limit

P At h = 1 the stabilizers drop out, and H(1, t) = − j ∈{a,b} (Xj + Zj ) + Kab with Kab = −(Xa + Za + Xb + / Zb ) − tQ. Every√spin other than a and b is in the single-spin ground state |q+ ⟩ = cos(π/8)|0⟩ + sin(π/8)|1⟩, with (X + Z)|q+ ⟩ = 2√ |q+ ⟩. The antisymmetric state of the contact pair has energy t, and in the symmetric basis {|00⟩, (|01⟩ + |10⟩)/ 2, |11⟩} √   0 −2√− t − 2 √ sym Kab = − 2 (F6) t − 2 , √ 0 − 2 2−t whose eigenvalues solve λ3 + tλ2 − (t2 + 8)λ − t3 = 0. With λ0 (t) the smallest root and |φ0 ⟩ its pair state, √ |Ω(1)⟩ = |φ0 ⟩ab ⊗ |q+ ⟩⊗(N −2) , E0 (1, t) = −(N − 2) 2 + λ0 (t),

(F7)

with λ0 = −3.1262 at t = 0.5. The ground state is a product state except for the contact pair, which is entangled for every t > 0.

Appendix G: Transferring angles between patch sizes

This appendix develops a method to transfer trained angles from small patches to larger ones, which avoids optimization at the target size. Such a method is needed because exact classical methods have limited reach, with our exact evaluator being able to train angles only up to N = 48. Transferring variational parameters between problem sizes has precedents in the quantum approximate optimization algorithm, where angles optimized on small graphs can remain effective on larger graphs with similar local structure [93–96]. That argument rests on a causal neighborhood of fixed depth, which a sequential FLDC does not have, so we motivate transfer instead by the locality of the ground-state response to the field (Figure 9(a)). Figure 9(b) shows the route we take: we first continue in h at a source size, then transfer the angles in N at fixed field. Appendix G 1 gives the locality argument, and App. G 2 tests a rule that matches local environments, which fails for generic patches, but works strips. We leave expansion to compact patches as future work.

1.

Why angles might transfer between sizes

The transfer rests on how the ground state responds to a change of the field. Let |Ω(h)⟩ be the nondegenerate ground state of H(h) with energy E0 (h), and |m(h)⟩ an excited eigenstate with energy Em (h). Differentiating the eigenvalue equation H|Ω⟩ = E0 |Ω⟩ with respect to h and projecting onto ⟨m| gives the first-order response ⟨m|∂h Ω⟩ = −

⟨m|∂h H|Ω⟩ . Em − E0

(G1)

Each excited state contributes with a weight inversely proportional to its excitation energy, so the state is most sensitive to the field where the gap γN (h) between the two lowest levels is small (Fig. 9(a)). A statement that holds uniformly in the size therefore needs bounded interactions and a gap that stays open along the whole path I = [0, h] for every patch size, γmin = inf inf γN (h′ ) > 0, ′ N h ∈I

(G2)

31 which gaps computed for finitely many patches cannot establish. Under this condition, quasi-adiabatic continuation [97, 98] replaces the sum over excited states in Eq. (G1) by a single operator. It constructs a Hermitian operator D(h), the quasi-adiabatic generator, whose commutator with the ground-state projector Π0 (h) = |Ω(h)⟩⟨Ω(h)| gives the change of the projector with the field, together with the unitary U(h) that this generator produces,   ∂h Π0 = i D, Π0 , ∂h U = iD U , U (0) = 11. (G3) Integrating this flow from h = 0 gives Π0 (h) = U(h)Π0 (0)U(h)† , so U(h) carries the exactly known zero-field ground state P to the ground state at field h. This is the dressingPthat the variational sweeps have to approximate. For H = j Hj and 0 < γ ≤ γmin , the generator is a sum D = j Dj of pieces, one for each term of the Hamiltonian, Z ∞ Dj (h) = dτ Wγ (τ ) eiH(h)τ ∂h Hj (h) e−iH(h)τ . (G4) −∞

Each piece takes the derivative ∂h Hj of one Hamiltonian term, evolves it in time under the full Hamiltonian, and averages the result over evolution times τ with a filter function Wγ . The filter is real and odd, decays rapidly for |τ | ≳ 1/γ, and is chosen so that its Fourier transform equals i/ω at every frequency at least as large as the gap, Z ∞ i cγ (ω) = W dτ Wγ (τ ) eiωτ = (|ω| ≥ γ). (G5) ω −∞ Between the ground state and an excited state, the time evolution in Eq. (G4) multiplies the matrix element by eiωτ with ω = Em − E0 ≥ γ, and the integral over τ then yields the factor of Eq. (G5), ⟨m|iD|Ω⟩ = −

⟨m|∂h H|Ω⟩ , Em − E0

(G6)

which is Eq. (G1). The generator therefore encodes the full response of the ground state, but, unlike the sum over excited states, it is built from the local terms of the Hamiltonian. The locality of the pieces follows from the Lieb–Robinson bound [99]. In a lattice system with bounded local interactions, a local operator evolved in time spreads at most with a finite velocity v, the Lieb–Robinson velocity. After a time τ , its weight outside a distance v|τ | from its initial support is exponentially small, although not zero, which defines an effective light cone (Fig. 10(a)). Because the filter suppresses evolution times beyond about 1/γ (Fig. 10(b)), each piece Dj only involves the evolution of ∂h Hj up to those times, and it is therefore concentrated within a distance (r) of order ξ ∼ v/γ of its term. To make this quantitative, let Aj (τ ) = eiHτ (∂h Hj )e−iHτ , and let Aj (τ ) be the same

E1 (N, h) U (h) γN (h) stabilizer state 0

N

(b)

small gap

exact angles at h = 0

(a) E

product state E0 (N, h) 1

h

continuation, source size

Ntar

(c)

transfer at fixed h

full star

used without refinement

truncated star plaquette window, r = 1

Nsrc

µ exact evaluator, N ≤ 48 0

htar

1

h

class of µ in a 3 × 3 patch

FIG. 9. Spectrum and angle preparation. (a) Sketch of the two lowest levels along the field. The ground energy E0 (N, h) connects the stabilizer state at h = 0 to the product state at h = 1, and the gap γN (h) to the first excited level is small in the region of the bulk transition (gray). Where the gap is open, the ground state is the image of the zero-field state under the quasi-adiabatic unitary U(h) of Eq. (G3). The curves place neither a transition nor a gap closing for a finite patch. (b) Routes to the angles of a target at (htar , Ntar ) in the plane of field h and size N . Exact angles are known at h = 0 for every size (green line), and optimization with the exact evaluator is possible for N ≤ 48 (blue region). We continue in h at a source size (blue arrow), transfer the angles in N at fixed field (purple arrow), and use them at the target without refinement (red circle). Continuation at the target size itself is possible only inside the blue region. (c) Class of a gate. The pair block µ acts on two spins on consecutive edges of a face (purple arc). Its class records its role in the sweeps and the labeled Hamiltonian terms in a square window of radius r around it (dashed), where a star truncated by the boundary differs from a full one. Gates of the same class, up to a symmetry of the square lattice, receive the circular mean of the source angles of that class, Eq. (G9).

32 (a)

τ

(b) (c)

|x| ≲ v|τ |

same labeled Hamiltonian in both windows

|Wγ (τ )|

ξ ∼ v/γ |τ | ≲ 1/γ x

tails decay beyond 1/γ but do not vanish

∂h Hj

Hk

source

target

(r) (r) Dj = Dk under the identification,

|τ | outside the cone the influence is suppressed, not zero

Hj

1/γ

the full pieces differ by their tails

FIG. 10. How a quasi-local generator arises and why it suggests transfer. (a) The Heisenberg evolution of a local term ∂h Hj spreads within a light cone |x| ≲ v|τ | (purple), with influence outside the cone that is exponentially suppressed but not zero, where the Lieb–Robinson velocity v is the largest speed at which local operators spread under the Hamiltonian. The filter of Eq. (G4) weights evolution times |τ | ≲ 1/γ (green band), so the piece Dj is concentrated within a distance ξ ∼ v/γ of the term (dark region). (b) Envelope of the filter, which decays beyond 1/γ without a sharp cutoff. The drawn envelope is not a specific filter, and v/γ is a crossover scale rather than a support radius. (c) Two terms in patches of different size whose windows of radius r carry identical labeled Hamiltonians have identical truncated pieces D(r) , Eq. (G7). Gates in identical windows are therefore candidates for receiving the same angle, which is the hypothesis behind the matching rule of App. G 2. (r)

evolution under the Hamiltonian restricted to a distance r beyond the support of Hj . Define Dj

to be the piece built

(r) from Aj with the same filter. Splitting the integral in Eq. (G4) at a cutoff time T > 0 gives (r)

Dj − D j

Z ≤

(r)

dτ |Wγ (τ )| Aj (τ ) − Aj (τ ) Z + 2 ∥∂h Hj ∥ dτ |Wγ (τ )|. |τ |≤T

(G7)

|τ |>T

The first term compares the full and the truncated evolution at short times, and the Lieb–Robinson bound makes their difference exponentially small as long as the light cone has not reached the truncation radius, T < r/v. The second (r) term collects the long times, which the decay of the filter suppresses. Choosing T just below r/v therefore makes Dj a good approximation of Dj , with an error that decays rapidly with r, but does not vanish [98]. This has a direct consequence for transfer. Two terms, possibly in patches of different size, whose neighborhoods of radius r carry identical labeled Hamiltonians along the whole path have identical truncated pieces D(r) (Fig. 10(c)). The quasi-adiabatic dressing near such terms is therefore the same up to these tails, whatever the size of the patch, which suggests that gates whose surroundings agree should receive similar angles. This motivates the transfer rules below, but is susceptible to errors when an optimizer does not select circuit coordinates that follow the local pieces. 2.

Transfer rules

We consider two rules that build the angles of a target patch from the best-found angles of smaller trained patches at the same field. Both give every dressing angle a class and assign the same angle to gates of the same class, but they define the classes differently and obtain the source angles differently. Classes. The role τ (µ) of a dressing angle µ records the prefix or suffix stage, the sweep, the slot of its pair within the face, and the rotation within the pair block, but not the position of the face. Its window of radius r ∈ {0.5, 1, 1.5, 2.5} is the square of half-width r lattice units centered on the gate, see Fig. 9(c). A class combines the role with the labeled Hamiltonian terms in the window, where a star truncated by the boundary differs from a full one. Rule 1 (environment matching). Rule 1 takes its source angles θbsrc from patches optimized independently, and it identifies two classes when they agree up to one of the eight symmetries of the square lattice. This identification ignores that a lattice symmetry need not preserve the order of the sweeps. Let Gr (µ) be the set of source angles in the class of a target angle µ, with resultant X 1 bsrc mµ = eiθµ′ , ρµ = |mµ |. (G8) |Gr (µ)| ′ µ ∈Gr (µ)

33 the same neighborhood lies in the bulk

(a)

(b)

source 1 × 3 E

M

E

M

M

M

E

M

the neighboring faces touch the boundary

E

? E

(c)

source 1 × 3

E

no source gate has two perpendicular neighbouring faces

target 1 × 5 target 2 × 2

source 3 × 3

target 5 × 5

FIG. 11. Coverage of a target by its sources. (a) The faces of a one-row strip are of two kinds, end faces E with one neighboring face and middle faces M with two. The best-found angles of a 1 × 3 source form a table with one entry per kind and role, from which a 1 × 5 target, or any longer strip, is filled (arrows). (b) Every face of a 2 × 2 patch has one neighbor beside it and one above or below it, an arrangement that never occurs in a one-row strip, so a strip source offers nothing to the gates of this target. (c) The highlighted faces are both called interior, but in the 3 × 3 patch the neighboring faces touch the boundary (dashed box), whereas in the 5 × 5 patch the same neighborhood lies in the bulk. The class of a gate compares the Hamiltonian terms in a window of stated radius together with its role, and the dashed boxes illustrate such a window.

The transferred angle is the circular mean θµtr = Arg mµ ,

(G9)

which respects the 2π periodicity of a rotation angle. Angles 0.02 and 2π − 0.02 have circular mean zero, whereas their arithmetic mean is π. A resultant length ρµ near one indicates tightly clustered source angles. A target angle without a match receives the circular mean over all source angles of the same role. The bridge and anchor angles carry no spatial environment, so each forms a class of its own and receives the circular mean over all sources. Nothing overrides the rule at h = 0, where we know analytic formulas. Coverage. A rule can only transfer classes that occur in the sources. For Rule 1, the coverage Cr =

#{µ in the target dressing : Gr (µ) ̸= ∅} Nptar − 2

(G10)

is the fraction of the Nptar − 2 dressing angles of the target, which exclude the bridge and anchor angles, whose class occurs in at least one source. The remaining fraction 1 − Cr falls back to a role mean. Coverage depends on which neighborhoods the sources contain, and it decreases with the window radius, because a larger window distinguishes more environments. Figure 11 illustrates this. A short strip covers every longer strip, since both contain only end faces and middle faces (panel a). A strip offers nothing to a compact patch, whose faces have neighbors in two directions (panel b). A face that lies in the interior of a small patch can still see the boundary within its window, unlike the same face in a larger patch (panel c). For the 3 × 3 target, all smaller patches together cover every angle at r = 0.5, thanks to the symmetry identification, but almost none at r ≥ 1.5. Full coverage, however, does not make Rule 1 accurate (App. G 3), so coverage is necessary but not sufficient. In particular, Rule 1 averages angles from patches that were optimized separately. Many different angle vectors prepare the same state, so each optimization settles on its own convention, and averaging across patches mixes conventions that do not fit together. Rule 2 avoids this by fixing one convention for all patches at once. Rule 2 (joint class table). Rule 2 builds a lookup table with one angle per class and trains this single table on several small patches together. The classes are those of Fig. 9(c) at the smallest window, r = 0.5, except that we keep track of orientation, that is, of the direction in which the pair block points and of which neighbors lie on which side. We do not merge a class with its rotated or mirrored copy, because in a sequential circuit these are different gates. We train the table by minimizing the average energy per spin of all source patches, starting from four random tables and keeping the best, since the zero-field angles form a stationary point from which the optimizer does not move. Training on all sources together forces each entry to work in every place where its class appears, rather than in one patch only. The trained table then fills the target directly, without any further optimization.

34

h = 0:1

(b)

continuation (D) transfer + refine (TD)

10 2

(a) excess energy per spin

10 2

h = 0:2

h = 0:25

(c) 10 2

transfer only (T) zero-field angles

10 0

10 0

10 0

10 ¡2

10 ¡2

10 ¡2

10 ¡4

10 ¡4

10 ¡4

10 ¡6

10

20

30

40

50

10 ¡6

10

20

N

30

40

10 ¡6

50

10

20

N

30

40

50

N

FIG. 12. Transferred angles against continuation. Excess energy per spin over the reference against N at (a) h = 0.1, (b) h = 0.2, and (c) h = 0.25. Blue, continuation in h at the target size. Magenta, transferred angles before any target update, with the matching rule at the radius that is best at each size and field. Purple, the same after refinement at the target size. Gray, dashed, the exact zero-field angles reused unchanged. Symbols mark the best and bars the worst of three seeds. At N = 38 and 44 only transferred and zero-field angles exist, and at N = 48 continuation exists at h = 0.1 and 0.25 and the refined transfer at h = 0.25.

(a)

(b) 0.6

10¡1

1.00

1.2

0.5

circular spread [rad]

within-class dispersion

excess / spin of T

100

N = 14 N = 26 N = 20 N = 32 ¡ 2 10 N = 24 N = 34 0.00 0.25 0.50 0.75 class coverage

(c)

0.4 0.3 0.2 0.1 0.0

0.25

0.50

h

0.75

1.00

1.0 0.8 0.6 0.4 0.2 0.0

median class 90th percentile

10

20

N

30

FIG. 13. Why the transfer fails, first campaign at h > 0. (a) Excess energy per spin of the transferred state against the class coverage of its prediction, one shade per target size. (b) Within-class circular dispersion 1 − ρ of directly optimized angles against h for the median class (solid) and the 90th percentile (dotted). (c) The corresponding circular spread against N . At N = 32 and 34 the spread is small, which we attribute to runs that hardly left their start. Panel (a) pools the eighteen fields with h > 0 of the first campaign.

3.

Performance of the transfer rules

We test both rules on targets that the exact evaluator can still handle, so that every transferred state can be compared with the best state found by continuation. The results follow the picture of App. G 2. Rule 1 does not carry the angles over, whereas Rule 2 transfers them accurately to strips, but not yet to compact patches. Rule 1. Figure 12 shows Rule 1 on targets of 14 to 48 spins. Before any update at the target, the transferred angles lie about 0.8 per spin above the ground energy, for every target, nonzero field, and window radius. Copying each angle from the gate at the same relative position in a source, which we call the position copy, behaves the same way. On an energy scale of about one per spin, this is close to a random state and further from the ground state than the zero-field angles reused unchanged, which stay within about 0.1 per spin up to h = 0.25 at every size up to N = 80 (App. F 3). Refining the transferred vector at the target recovers most of this gap, but it ends at or above continuation there and is no cheaper from N = 26 on. The diagnostics below show that the obstacle is not missing coverage but the mismatch of conventions anticipated in App. G 2. Why Rule 1 does not transfer. The step from Eq. (G3) to circuit angles is not a theorem, and the diagnostics show

35

¡0.6 ¡0.8 ¡1.0

1£ 20 3 2£ 24 2 1£ 26 4 1£ 32 5 2£ 34 3 1£ 38 6 2£ 44 4

¡1.2

ground state zero-field angles direct optimization from Z direct optimization from random

target: N, patch

10¡1

zero-field angles

10¡2 10¡3

38 (1 £ 6) 32 (1 £ 5)

continuation in h transferred transferred, then optimized

10¡4 0.0

0.5

1.0

share of classes set by one source gate

1.5

(c) 104 103 102 101 100 10¡1

1£ 20 3 2£ 24 2 1£ 26 4 1£ 32 5 2£ 34 3 1£ 38 6 2£ 44 4

¡0.4

20 (1 £ 3) 44 (2 £ 4) 34 (2 £ 3) 26 (1 £ 4)

seconds at the target

(b)100

excess per spin, untouched transfer

E=N at h = 0:25

(a) ¡0.2

target: N, patch

FIG. 14. Joint class table against optimization at the target, at h = 0.25. In (a) and (c) each column is one target patch, with compact patches shaded. (a) Energy per spin of every route, with the reference (black bar) and the zero-field angles (gray). Blue, direct optimization at the target with a budget of 100 iterations or 900 s, from the zero-field angles (open) and from random angles (filled, mean and range over 10 draws). Green, continuation in h at the target. Magenta, the joint class table expanded onto the target without optimization, and purple, the same refined with the direct budget. (b) Excess energy per spin of the untouched transfer against the share of target angles whose class a single source gate decides, filled for strips. The gray band marks the excess of the zero-field angles. (c) Seconds spent at the target for the three routes that start from smaller patches. The one-off training of the table, which serves every target of a family, is not included.

where it breaks (Fig. 13(a)). On the 3 × 3 target at h = 0.1 to 0.3 the fully covered rules leave 0.53 to 0.71 per spin and the almost uncovered ones, which fall back to role means, 0.29 to 0.38. The resultant length explains why. Among directly optimized angles of one class the median resultant length is 0.81, a circular spread of 0.65 rad, and for one class in ten it lies below 0.51, a spread above 1.16 rad, see Fig. 13(b,c). A class mean therefore shifts every gate by several tenths of a radian, enough to change a state built from four sweeps of such gates substantially. Tying all gates of a class to one shared angle leaves a reduced circuit of 122 free angles instead of 642, and training it directly at N = 24 leaves 0.10 per spin at h = 0.25 and 0.13 at h = 0.3, against 10−4 per spin for the free optimization at the same size and budget. The tested optimizations did not find an accurate vector within these symmetrized classes at the available budget, which does not exclude more accurate vectors or better optimizers in the same parametrization, and the oriented classes of Rule 2 do reach good accuracy. The other expectations behind Rule 1 are not borne out either. Complete coverage does not improve the transferred energy (0.87 against 0.78 per spin at partial coverage), the transfer is least accurate in the polarized limit (1.43 per spin) rather than in the transition window (0.72), and the trained angles are not smooth within classes. Rule 2. Rule 2 needs more than coverage. Every table entry that the target uses must have been trained on at least two gates in the sources. An entry trained on a single gate simply inherits the convention that one optimization happened to pick, which is the problem of Rule 1 again. An entry shared by several gates, in patches of different sizes, has to work for all of them, and this pushes it toward the local dressing suggested by Eq. (G3). Like coverage, the condition can be checked by counting gates, before any training. Strips pass it easily, because middle faces appear in every strip source, so strip sources of up to four plaquettes suffice for every strip of five or more. On strips, Rule 2 transfers accurately (Fig. 14). On the 1 × 5 strip at h = 0.25, with shorter strips as sources, the transferred angles lie 1.2 × 10−3 per spin above the reference without any update, at least an order of magnitude closer than direct optimization at the target, and only continuation in h does better, at 2.6 × 10−4 . The transfer needs a single energy evaluation at the target, after an eight-minute training of the table that serves every longer strip, whereas continuation takes about three hours per field sweep. At N = 38, beyond our continuation runs, the table still leaves 1.3 × 10−3 per spin, against 0.115 for direct optimization. On every tested target, the transfer is accurate exactly where the condition above holds (Fig. 14(b)), which we use as an empirical criterion rather than a guarantee. Compact patches do not yet meet the condition, because each of our sources, of at most 34 spins, contains their interior faces only once, so for them transfer remains to be demonstrated. Adding the 2 × 4 patch as a source would meet it for the 3 × 3 target, although it covers only 89% of its angles, and we have not tested it. The compact patches of the device runs, from 4 × 4 on, would need a source with two interior faces, the 3 × 4 patch with 62 spins, which is beyond our exact evaluator.

36 Appendix H: Knitted estimators and seam entropy

This appendix gives the knitted estimators of the energy and the contact correlation, the bridge-angle dependence of the energy, and the seam entropy of the prepared states. The reconstruction uses the local-operation protocol and estimator framework of Refs. [14, 15].

1.

Energy and contact correlation

A setting (i, p, g) runs branch i of Fig. 4(c), with weight wi ∈ {c2 , s2 , d, −d, d, −d}, on patch p with readout basis (i) g ∈ {X, Z}. For a Hamiltonian Pauli term Pν of group g, let ȳν,p be the mean over shots of the product of its measured spin values on patch p, multiplied by the recorded sign if branch i measures on patch p. Empty products equal one, so a term on one patch picks up the mean sign of the branch on the other. The energy estimate is b= E

6 X i=1

wi

X

(i)

(i)

Jν ȳν,A ȳν,B ,

(H1)

ν

which is unbiased for the ideal circuit because the two patches are sampled independently and every branch state is a product across the seam. Each of the 24 settings first receives 32 shots, and the rest of M = 24,576 is allocated in proportion to |wi | times the estimated standard deviation of the setting. Error bars come from bootstrap resamples of whole bit strings within each setting. Contact correlation. We reconstruct ⟨Za Zb ⟩, ⟨Za ⟩, and ⟨Zb ⟩ from the all-Z branch records using Eq. (H1), with the Hamiltonian replaced by the corresponding observable. We then form Cab = ⟨Za Zb ⟩ − ⟨Za ⟩⟨Zb ⟩, rather than combining branchwise connected correlations. For each bootstrap replicate, the same resampled records are used for all three expectations before forming Cab , preserving their statistical dependence. The three expectations are unbiased for the ideal circuit, but the product ⟨Za ⟩⟨Zb ⟩ makes the finite-shot estimate of Cab nonlinear, so we do not claim it is unbiased. Applied to this model, the sampling bound of Eq. (3) gives a size-independent guarantee. For an estimator that samples one branch with probability |wi |/Γ and one readout group in proportion to its Hamiltonian weight, each observation of E/N is bounded by Γ[1 + h + (2(1 − h) + t)/N ], which does not grow with N . Explicitly, the total weight of the Hamiltonian is X W = |Jν | = (1 − h)(N + 2) + 2hN + t, (H2) ν

each observation of E/N lies in [−AN , AN ] with AN = ΓW/N , and Eq. (3) gives h M ε2 i   Pr |e − E/N | ≥ ε ≤ 2 exp − 2 , 2AN

(H3)

with AN ≈ 1.3 at h = 0, several times the measured error constants of Fig. 5(a).

2.

Bridge angle

This subsection shows that the same measurement outcomes can be used to reconstruct the energy at any bridge angle, and asks whether the reconstruction is precise enough to locate the best bridge angle and thus resolve what the bridge contributes as the patches grow. In the uncut circuit, the bridge angle ϕ must be fixed before the bridge gate is applied. After cutting, the six branch circuits contain only the patch angles and fixed local operations, and ϕ enters only through the weights w(ϕ) = (c2 , s2 , d, −d, d, −d) of Eq. (H1). Running each branch circuit on both patches gives its unweighted contribution P (i) (i) E (i) = ν Jν ȳν,A ȳν,B , including any recorded measurement signs. Reweighting these six numbers gives the energy at any bridge angle, for fixed patch angles. Since c2 = (1 + cos ϕ)/2, s2 = (1 − cos ϕ)/2, and d = 21 sin ϕ, E(ϕ) =

6 X i=1

wi (ϕ) E (i) = e0 + e1 cos ϕ + e2 sin ϕ,

(H4)

37 with e0 = 12 (E (1) + E (2) ), e1 = 12 (E (1) − E (2) ), and e2 = 12 (E (3) − E (4) + E (5) − E (6) ). For nonzero amplitude, e21 + e22 > 0, its minimum lies at ϕ⋆ = atan2(−e2 , −e1 ),

E(ϕ⋆ ) = e0 −

q e21 + e22 .

(H5)

If e1 = e2 = 0, every angle minimizes E, and from sampled records we write eb1 , eb2 , and ϕb⋆ for the estimates. Once records covering all six branches are available, thus determining e1 and e2 , we can therefore derive the best bridge angle for the chosen patch angles, without new angle-dependent circuits, to a precision set by the number of shots. We test this precision by sampling the ideal branch circuits with the shot budget of the device runs. At N = 34, the reconstructed curves (solid) follow the exact (dashed) E(ϕ) of the same patch angles at every field h, but their minima are not a reliable guide to the best angle (Fig. 15(a)). Minimizing a noisy curve favors angles at which the noise lowers the estimate, so the reconstructed minima lie up to two standard errors, about 0.01 per spin, below the exact energy at the selected angle. That exact energy stays within 4 × 10−3 per spin of the best value along ϕ, so the apparent gain from re-optimizing the bridge is statistical rather than real. As the patches grow, the precision of the reconstruction degrades even though the effect of the bridge does not. The zero-field circuit makes this visible, because its landscape is known exactly at every size. With no dressing and α = π/2, App. F gives E(ϕ) = −N + 2 − Dt cos(ϕ − ϕ0 ),

Dt =

p 16 + t2 ,

ϕ0 = arctan(t/4),

(H6)

so the offset e0 = −N + 2 grows with the size, whereas the amplitude Dt and the optimum ϕ0 do not. The bridge itself lowers the energy by only Dt − 4 ≈ 0.031 relative to ϕ = 0, a small fraction of the oscillation amplitude Dt ≈ 4.03. After subtracting e0 , the exact curves of all sizes coincide and share the optimum ϕ0 (Fig. 15(b,c)). The reconstructed curves in Fig. 15(b) scatter around this common curve, with bands that widen as N grows. Likewise, the estimated minimizers ϕ⋆ of Eq. (H5) scatter around ϕ0 more strongly at larger sizes (Fig. 15(c)). Scanning the bridge angle is therefore free, but it cannot improve the bridge, already at N = 34 and increasingly at larger sizes, due to statistical fluctuations in the angle-dependent coefficients eb1 and eb2 . The bridge changes the energy by an amount that does not grow with the size, and which the extensive noise of the full energy hides. This is why Sec. IV D measures the contact correlation, which isolates the interface, rather than relying on the energy.

h=0 h = 0:1 h = 0:2 h = 0:25

¡0.25 ¡0.50 ¡0.75 ¡1.00 ¡1.25

h = 0:3 h = 0:5 h=1

(b)

h=0

10

N = 48 N = 62

(c) N = 98 N = 142

5 0

¡5 ¡¼

0 bridge angle Á

¼

¡¼

0 bridge angle Á

re-optimized angle Á ⋆ at h = 0

N = 34

E(Á) ¡ e0

E(Á)=N

(a)

¼

exact sampling stabilizer sampling

1.0 0.5

arctan(t=4)

0.0 8

24 48 62 98 142 N

FIG. 15. Knitted records give the energy at every bridge angle, but shot noise hides the effect of the bridge as the patches grow. Noiseless simulations of the ideal branch circuits at the shot budget of the device runs, with the patch angles fixed and the same six-branch records reweighted for every bridge angle ϕ, at t = 0.5. (a) Energy per spin E(ϕ)/N at N = 34, one shade per field, reconstructed from exact state-vector samples (solid, with bootstrap bands) and evaluated exactly for the same patch angles (dashed). Circles mark the trained angles, stars the reconstructed minima, and open diamonds the exact energies at the reconstructed minimizers. (b) Angle-dependent part E(ϕ) − e0 at h = 0 for N = 48, 62, 98, and 142, reconstructed from √ stabilizer samples of the Clifford zero-field branch circuits (App. F). The exact curve, −Dt cos(ϕ − ϕ0 ) with Dt = 16 + t2 , is the same at every size, while the bands of the reconstructions widen with N . (c) Estimated minimizer ϕ⋆ at h = 0 against N , from exact samples for N ≤ 34 (open circles) and stabilizer samples beyond (squares), with the exact optimum ϕ0 = arctan(t/4) (dashed). Bands and error bars are bootstrap uncertainties.

38 3.

Seam entropy

For a single severed gate, the seam entropy follows exactly from the states of the two patches just before the bridge. Proposition 7 (Seam entropy of a single bridge). Let |uA ⟩|uB ⟩ be a product state, let za = ⟨uA |Za |uA ⟩ and zb = ⟨uB |Zb |uB ⟩, and let ab |ψ⟩ = (WA ⊗ WB )RZZ (ϕ)|uA ⟩|uB ⟩

(H7)

for local unitaries WA and WB . Then the at most two nonzero eigenvalues of ρA satisfy λ+ + λ− = 1,

λ+ λ− = c2 s2 (1 − za2 )(1 − zb2 ),

(H8)

and q √
2 Γsep (ψ) = 2 Tr ρA − 1 = 1 + 2| sin ϕ| (1 − za2 )(1 − zb2 ) ≤ Γ(ϕ).

(H9)

√ Equivalently, the seam entropy is S1/2 = 2 log2 Tr ρA . Proof. Local unitaries WA and WB do not change the spectrum of ρA , so we set them to the identity. For each patch, q Zbp |up ⟩ = zp |up ⟩ + 1 − zp2 |u⊥ (H10) p ⟩, ⊥ where |u⊥ p ⟩ is normalized and orthogonal to |up ⟩. In the orthonormal product basis built from |up ⟩ and |up ⟩, the coefficient matrix of |ψ⟩ is p   c −p isza zb −isza 1 − zb2 p C= , (H11) −iszb 1 − za2 −is (1 − za2 )(1 − zb2 )

with determinant q det C = −ics (1 − za2 )(1 − zb2 ). The nonzero eigenvalues of ρA = CC † have sum Tr CC † = 1 and product | det C|2 , giving Eq. (H8). Since p p
2 p λ+ + λ− = 1 + 2 λ+ λ−

(H12)

(H13)

√ and 2|cs| = | sin ϕ|, Eq. (H9) follows from the pure-state formula Γsep (ψ) = 2(Tr ρA )2 − 1 [48]. The result extends continuously when one of the two local spans becomes one-dimensional. In the simulations, za and zb are estimated by sampling prefix circuits, which run the encoder and the prefix sweeps of one patch and measure its bridge spin. For the ground state, any state of Schmidt rank two across the seam satisfies (0)

(0)

|⟨Ω|ψ⟩|2 ≤ λ1 + λ2 , (0)

(H14)

(0)

where λ1 ≥ λ2 are the two largest Schmidt weights of the exact ground state |Ω⟩, because such a state occupies a two-dimensional subspace of patch A. For N ≤ 24, the weight beyond these two stays below 10−6 at every field, so a single bridge does not limit the fidelity.

Appendix I: Gradient-variance diagnostic

The trainability guarantee for FLDCs [41] assumes a circuit of local blocks with bounded support β, each spin acted on by a bounded number χ of blocks, and every block drawn independently from a local 2-design. For a cost made of local terms, it then bounds the gradient variance from below independently of the system size. However, the gradient variance of a single local observable still decays exponentially with its causal distance ∆k to the parameter, the number of blocks on the shortest chain that carries the effect of the parameter forward to the observable (Fig. 16(a)), but not with the total size. Our circuit has bounded support, β = 2, bounded local depth (Sec. IV A), and a cost made of local terms, but its pair blocks, built from fixed rotations with independent angles, are not drawn from a 2-design. This

39

(a)

cost: the single star Afar

(b)

¢k = 1 ¢k = 2 ¢k = 3

Varµ [∂¹ C]

T

¢k = 4 ¢k = 5

6 £ 10¡3 4 £ 10¡3 3 £ 10¡3

¢k = 3 time

2 £ 10¡3

both costs, one circuit no gate is farther from its nearest term

10¡4 Varµ [∂¹ C]

10¡2

µ¹

(c) 10¡2 10¡6 10¡8

total energy, N = 20 total energy, N = 26 total energy, N = 32 total energy, N = 38 single star Afar , N = 32

10¡10 10¡12

10

20

30

10¡14

0

5

N

10 15 20 causal distance ¢k

FIG. 16. Gradient variance at random initialization (T3). (a) A sequential sweep. Blocks (boxes) act on pairs of neighboring spins (lines) in turn, and time runs to the right. The number of blocks acting on each spin, given at the right, stays at most two however many spins the staircase covers, which is the finite local depth of an FLDC. The causal distance ∆k from the parameter θµ (purple) to the measured term T (green) counts the steps of the shortest chain of blocks to the last block on the support of T , here ∆k = 3, and it grows with the number of spins. (b) Variance of the gradient of a single star Afar at h = 0.25 over uniformly random angles against N , at fixed ∆k (light to dark), for strips. Afar is the weight-two star of patch A farthest from the contact, at the open end of the strip. (c) The same variance against ∆k, for Afar at N = 32 (black) and for the total energy at N = 20, 26, 32, and 38 (magenta, light to dark), where ∆k is the distance to the nearest Hamiltonian term. The dotted line marks the largest such distance of any gate. Bars are 95% bootstrap intervals.

Distance from the nearer end

N =8

14

20

26

32

38

0 0.5 1 1.5 2 2.5

69.0 – – – – –

21.8 20.9 – – – –

17.7 13.6 8.95 – – –

17.7 13.0 7.41 4.42 – –

17.7 13.2 7.52 4.16 3.34 –

18.4 13.4 7.49 4.02 3.12 2.73

TABLE VI. Variance of the energy gradient on strips at h = 0.25, averaged over the gates at a given distance from the nearer end of the strip, in plaquettes, and given in units of 10−3 . The entries of a row agree wherever the strip is long enough to contain that distance.

appendix therefore checks the two consequences directly, that a single observable in our circuit follows the same trend, and additionally confirms that the total energy, the cost we actually train, does not decay with ∆k at all. Setup. We use strips with N = 8 to 38 and compact patches with N = 24 to 48. For each random draw of the angles, the exact evaluator of Sec. IV B returns the full gradient in one reverse-mode pass. The number of draws falls from 2,000 at N ≤ 20 to 64 at N = 48, because each draw requires one exact gradient, whose cost grows exponentially with the patch size. The variance of each partial derivative is taken over the draws. All runs use h = 0.25 and t = 0.5, and we consider the total energy E rather than E/N , as in the main text, to align with Ref. [41]. We draw every angle independently and uniformly from [0, 2π), the natural random initialization of our circuit. We additionally surround each probed RZZ rotation with Haar-random two-qubit unitaries, to mimic the FLDC local 2-design assumption. The variance then falls faster, from 7.6 × 10−2 at N = 8 to 1.2 × 10−3 at N = 26, indicating that the weak size dependence we report is therefore specific to the natural initialization of our circuit, not a property of random blocks in general. A single observable. We first take the weight-two star Afar of patch A farthest from the contact. Its causal distance grows by three with every added plaquette, from 10 at N = 8 to 25 at N = 38. At fixed ∆k, its gradient variance does not depend on the length of the strip, once the strip is longer than one plaquette (Fig. 16(b)). Along ∆k, it falls by a factor of about 0.28 per block at N = 32 (Fig. 16(c)), which is the same behavior found for FLDCs in Ref. [41]. P The total energy. The energy gradient is a sum over local terms, ∂µ E = a ∂µ ⟨Aa ⟩, and every parameter has some term within eight blocks on strips and nine on compact patches. Unlike that of the single star, its variance shows no decay with ∆k (Fig. 16(c)). We attribute this to the nearby terms dominating the sum, so that the energy does not inherit the long-distance decay of a single far observable.

40 Size dependence. The average energy-gradient variance decreases with size, but not exponentially, because the decrease comes from the open ends of the patches, which have limited impact at larger sizes. We locate each gate by its distance, in plaquettes, from the closer end within a strip. Table VI lists the variance averaged over the gates at each such distance, with one column per strip length. Along a row, the variance of a gate depends on its position and not on the length of the strip, assuming a long enough strip to contain that distance. Down a column, it falls from about 1.8 × 10−2 at the ends to about 2.7 × 10−3 two and a half plaquettes inside. A longer strip therefore only adds gates with this bulk value, and its average levels off instead of decaying exponentially. Compact patches, however, are bounded on all sides. Their variance also falls away from the boundary, but it has not reached a bulk value at the sizes we can simulate, so whether it levels off remains open (Fig. 5(b)). For contrast, a control circuit that is identical except that its number of sweeps grows with the linear size of the patch has local depth 13 at N = 8 and 82 at N = 32. On strips from N = 8 to 32, a fit of ln Var = c − rN gives r = 0.233 ± 0.004 per spin, so its variance falls by about a fifth with every added spin.

Appendix J: Hardware circuits and settings

This appendix gives the details of the device runs that are not part of the knitting protocol, namely the circuit sizes and trained angles, the compilation and error suppression, and the empirical correction calibrated at h = 0 together with the depth dependence of the signal it measures.

1.

Circuit coordinates and trained angles

The angles of a circuit are stored as one vector, ordered as the prefix sweeps of patches A and B, the bridge angle ϕ and the anchor angle α, and the suffix sweeps of patches A and B. With spre prefix and spost suffix sweeps, its length is Np = 40 nx ny (spre + spost ) + 2,

(J1)

and we use spre = spost = 2. Table VII lists every patch geometry, with the two-qubit gate count and depth of the larger subcircuit, maximized over its branch circuits. The circuit structure is the same at every field, so we train by continuation in h. Starting from the exact zero-field angles and from an optimized h = 1 state, L-BFGS with exact gradients walks the field grid, starting each field from the previous one and keeping the lower energy. Up to N = 34, with three to five seeds, the best vector lies within two-qubit gates N

(nx , ny )

n

n+1

Np

D0

unrouted

routed

routed depth

8 14 20 24 26 32 34 48 62 80 98 120 142 168

(1, 1) (1, 2) (1, 3) (2, 2) (1, 4) (1, 5) (2, 3) (3, 3) (3, 4) (4, 4) (4, 5) (5, 5) (5, 6) (6, 6)

4 7 10 12 13 16 17 24 31 40 49 60 71 84

5 8 11 13 14 17 18 25 32 41 50 61 72 85

162 322 482 642 642 802 962 1,442 1,922 2,562 3,202 4,002 4,802 5,762

2 2 2 4 2 2 4 8 8 16 16 32 32 64

38 74 110 145 146 182 216 322 428 569 710 886 1,062 1,273

38 83 134 173 181 227 258 382 512 1,375 1,519 2,436 2,424 2,995

36 56 66 103 77 87 105 124 142 451 406 556 542 574

TABLE VII. Patch geometries of the studies and of the hardware sweep. The subcircuit width n + 1 includes the readout ancilla, which carries no angle, the number of stored angles Np follows from Eq. (J1) with two prefix and two suffix sweeps, and D0 = 2min(nx ,ny ) is the exact MPS bond dimension of the zero-field state. The last three columns give the native two-qubit gate count of the largest branch circuit of the series in Fig. 6, before and after layout and routing on ibm phoenix, and its two-qubit depth after routing.

41 2 × 10−3 per spin of the reference at every field. At N = 48, two-hour runs allow only about 110 gradient evaluations for 1,442 parameters and barely improve on the zero-field angles. The first hardware series of Sec. IV D uses the best trained vectors.

2.

Compilation and error suppression

All device data are taken on ibm phoenix. Every branch circuit is compiled with Qiskit [100] against the current backend target, using the SWAP-based heuristic search SABRE [101] at optimization level 3. The circuits of each size are transpiled once with all rotation angles left as parameters, so the angles enter only as virtual Z rotations and the physical circuit is the same at every field. Idle qubits are protected by dynamical decoupling with pairs of X pulses of opposite phase [44], and every two-qubit gate is Pauli twirled [45]. Every shot is post-selected with the inversion test of Ref. [46, App. H], implemented with qiskit-addon-utils [102]. After the final measurement, each checked qubit is flipped by an X pulse and measured again, and a shot is kept only if the two outcomes disagree on every checked qubit, which removes shots with readout or relaxation errors at the end of the circuit. The checked qubits are those of the subcircuit and the idle device qubits next to them. Increased error rates due to larger sizes and more checks make the median kept fraction of shots fall from 0.83 at N = 8 to 0.04 at N = 142. Fewer kept outcomes therefore enter each estimate at larger sizes, leading to increased statistical errors. Each point of a sweep uses M = 24,576 shots for the 24 branch settings and 2,048 shots for one prefix circuit per patch. The sixteen series shown, about 3 × 106 shots in total, used 317 seconds of charged processor time, about twice the execution time because of twirling and per-circuit overheads.

3.

Signal loss and its depth dependence

The dominant effect of the device noise is a damping of expectation values toward zero. At h = 0, every submitted angle vector prepares the exact zero-field state, whose energy Eexact (0) is known (App. F), so the fraction of the signal that a series s retains is λs =

Es (0) . Eexact (0)

(J2)

A series is characterized by one size, one setting of the error suppression, and one choice of angles. Figure 17 shows the retained signal of every series. It falls from 0.82 at N = 8 to 0.53 at N = 34 for the trained angles (blue), and from 0.44 at N = 48 to 0.039 at N = 142 for the zero-field angles (black), with the sharpest drop between the 3 × 4

signal kept [%]

100 50

0 0.1 0.2

20 10 5 2 1 0.5 0.2 0.1

h

0.25 0.3 0.5

mean over h; bands §1 SE direct optimization zero-field angles transferred, then optimized

8

34

62

N

98

142

FIG. 17. Fraction of the signal kept at h = 0, the measured over the exact energy, against N for each kind of angle vector, with dynamical decoupling and post-selection. The correction of App. J 4 divides every energy of a series by this fraction.

42 and 4 × 4 patches. Two additional series (purple) with transferred and refined angles at N = 48 and 62 retain 0.38 and 0.35, like the zero-field circuits of the same sizes. The retained signal follows the routed two-qubit depth D of the larger subcircuit (Table VII) consistent with decoherence of idle qubits during the sequential sweeps. A weighted fit of − ln λs through the origin, over the fifteen series with λs > 0.02, gives λs ≃ e−cDs ,

c = (0.61 ± 0.03)% per two-qubit layer,

(J3)

so every layer of two-qubit gates removes about 0.6% of the signal.

4.

Correction calibrated at zero field

Since the h = 0 circuit has a known ideal energy, it can serve as a calibration circuit for the other fields of the same series. Dividing the energies of a series s by its retained signal λs measured at h = 0 (App. J 3) gives corrected bs (h) = Es (h)/λs , with the error of λs propagated, where we omit series with λs < 0.005. This rescaling energies E intuition rests on two facts. The physical circuit is the same at every field (App. J 2), and twirling makes the gate noise a Pauli channel on average. If that channel acted only after the last gate, each energy term would be damped by a factor independent of the angles, and the factor measured at h = 0 would correct the whole series. In practice, errors occur throughout the circuit and remain Pauli errors under propagation only in Clifford circuits, as at h = 0. The calibration also uses only star and plaquette terms, since the field terms vanish at h = 0, whereas single-spin terms are usually damped less. The rescaling is therefore a one-parameter approximation, not an error-mitigation guarantee. It cannot remove preparation error, and it multiplies statistical errors by 1/λs . For the trained angles at N = 8 to 34, the ideal energy is close to the ground energy, so the correction can be tested directly (Fig. 18(a,b)). It removes about 86% of the raw gap on average and brings the energies within 0.13 per spin of the reference for 0 < h < 1. At h = 1, where single-spin terms dominate, it overcompensates by about 0.15 per spin, below the reference. For the zero-field angles, the corrected energies should reproduce Eq. (F5) at every field (Fig. 18(c)). They do so within 2.3 standard errors up to N = 120, deviate by up to four at N = 142.

E=N

(a) 0.00

¡0.25 ¡0.50 ¡0.75 ¡1.00 ¡1.25 ¡1.50 ¡1.75

(b)

(c) 0.00

¡0.8 ¡1.0 ¡1.2 ¡1.4

raw rescaled at h = 0 DMRG

0.0

0.5

h

N = 14 N = 20 N = 32

¡1.6 1.0

0.0

227 CZ 83 CZ

0.5

h

1.0

¡0.25 ¡0.50 ¡0.75 ¡1.00 ¡1.25 ¡1.50 ¡1.75

2,424 CZ

512 CZ

N = 62 N = 98 N = 142 0.0

0.2

0.4

h

FIG. 18. Device energies corrected at h = 0. (a) Effect of the correction for N = 14, 20, and 32 with the directly optimized angles. Open markers joined by dashed lines are the raw device energies, filled markers joined by solid lines the same series divided by the fraction of the signal that its h = 0 point kept, and arrows join the two at every field. (b) The corrected series of (a), enlarged, with their error bars. (c) Corrected energies of the zero-field angles at N = 62, 98, and 142 for h ≤ 0.5. Thin dotted and dashed lines are the DMRG reference, which the zero-field angles do not reach for h > 0, so the relevant comparison in (c) is with their closed-form energy, Eq. (F5). Labels give the largest native two-qubit gate count of one routed subcircuit. At h = 0 the corrected point equals the exact energy by construction, and in (a) the sizes are shifted slightly in h.

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