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Beyond-classical computation in quantum simulation

Andrew D. King, Alberto Nocera, Marek M. Rams, Jacek Dziarmaga, Roeland Wiersema, William Bernoudy, Jack Raymond, Nitin Kaushal, Niclas Heinsdorf, Richard Harris, Kelly Boothby, Fabio Altomare, Mohsen Asad, Andrew J. Berkley, Martin Boschnak, Kevin Chern, Holly Christiani, Samantha Cibere, Jake Connor, Martin H. Dehn, Rahul Deshpande, Sara Ejtemaee, Pau Farré, Kelsey Hamer, Emile Hoskinson, Shuiyuan Huang, Mark W. Johnson, Samuel Kortas, Eric Ladizinsky, Tony Lai, Trevor Lanting, Ryan Li, Allison J. R. MacDonald, Gaelen Marsden, Catherine C. McGeoch, Reza Molavi, Richard Neufeld, Mana Norouzpour, Travis Oh, Joel Pasvolsky, Patrick Poitras, Gabriel Poulin-Lamarre, Thomas Prescott, Mauricio Reis, Chris Rich, Mohammad Samani, Benjamin Sheldan, Anatoly Smirnov, Edward Sterpka, Berta Trullas Clavera, Nicholas Tsai, Mark Volkmann, Alexander Whiticar, Jed D. Whittaker, Warren Wilkinson, Jason Yao, T. J. Yi, Anders W. Sandvik, Gonzalo Alvarez, Roger G. Melko, Juan Carrasquilla, Marcel Franz, Mohammad H. Amin

arXiv:2403.00910v2quant-phcond-mat.dis-nncond-mat.stat-mech

TL;DR

The paper asks whether beyond-classical computation can be demonstrated for practically meaningful quantum simulations. It studies quenched TFIM spin-glass dynamics with superconducting quantum annealers and classical approximations, finding QPU-quality behavior where leading classical methods become prohibitively costly.

  • Problem

    The central gap is limited evidence that near-term quantum processors outperform classical methods on problems of practical interest.

  • Method

    The paper simulates quenched TFIM spin-glass dynamics with superconducting quantum annealers and compares QPU-quality sampling against MPS, PEPS, and NQS approximations.

  • Results

    A few hundred qubits would imply a hypothetical Frontier runtime exceeding millions of years for MPS, with infeasible memory and energy requirements.

  • Takeaways & Limitations

    The results support beyond-classical quantum simulation for square, cubic, diamond, and biclique spin-glass topologies relevant to materials science and AI.

  • Takeaways & Limitations

    The classical-cost extrapolation relies on perfect parallelization and extends beyond the directly ground-truth-verifiable scale.

Abstract

from arXiv · show

Quantum computers hold the promise of solving certain problems that lie beyond the reach of conventional computers. However, establishing this capability, especially for impactful and meaningful problems, remains a central challenge. Here, we show that superconducting quantum annealing processors can rapidly generate samples in close agreement with solutions of the Schrödinger equation. We demonstrate area-law scaling of entanglement in the model quench dynamics of two-, three-, and infinite-dimensional spin glasses, supporting the observed stretched-exponential scaling of effort for matrix-product-state approaches. We show that several leading approximate methods based on tensor networks and neural networks cannot achieve the same accuracy as the quantum annealer within a reasonable time frame. Thus, quantum annealers can answer questions of practical importance that may remain out of reach for classical computation.

INTRODUCTION

The paper addresses whether beyond-classical computation has been established for practically meaningful problems, studying continuous-time TFIM dynamics with superconducting quantum annealers and classical approximations. It evaluates QPU-quality sampling against ground truths and extrapolates classical costs using entanglement scaling.

  • INTRODUCTION: Beyond-classical computation has mostly been demonstrated for random-number generation, while practical problems remain insufficiently established.
  • INTRODUCTION: The paper studies continuous-time transverse-field Ising-model dynamics using superconducting quantum annealing processors.
  • INTRODUCTION: Quantum critical dynamics are relevant to condensed matter physics and optimization, motivating simulations across programmable topologies of varying dimension.
  • INTRODUCTION: Small-scale QPU errors are estimated against converged MPS ground truths, while tensor-network and neural-network methods are assessed for matching QPU accuracy.
  • INTRODUCTION: Beyond the classically simulable regime, QPU results are extrapolated using observed and expected area-law scaling and universal quantum-critical scaling.

QUENCHING A QUANTUM SPIN GLASS

The model quench evolves a random-topology Ising spin glass from a paramagnetic state into a spin-glass phase by varying transverse and longitudinal energy scales. The transition crosses a topology-dependent quantum phase transition.

  • QUENCHING A QUANTUM SPIN GLASS: The time-dependent Hamiltonian interpolates between a driving Hamiltonian and a classical Ising problem Hamiltonian.
  • QUENCHING A QUANTUM SPIN GLASS: The evolution starts with Γ(0) ≫ J(0) in a paramagnetic phase and ends with Γ(1) ≪ J(1) deep in the spin-glass phase.
  • QUENCHING A QUANTUM SPIN GLASS: Random nonzero couplings define square, dimerized cubic, diamond, and dimerized biclique topologies separated by a topology-dependent quantum phase transition.
  • QUENCHING A QUANTUM SPIN GLASS: Two distinct QPU generations simulate the same model quench, with each processor generating at least 1000 samples per second.

CLASSICAL SIMULATION TECHNIQUES

The study compares QPU sampling with leading approximate solvers for time-dependent Schrödinger dynamics. MPS leverages area-law structure, whereas PEPS and NQS face contraction, truncation, locality, or sampling-cost challenges.

  • CLASSICAL SIMULATION TECHNIQUES: The comparison uses MPS, PEPS, and NQS as leading approximate frameworks for time-dependent Schrödinger-equation solutions.
  • CLASSICAL SIMULATION TECHNIQUES: MPS reduces representation complexity by exploiting area-law entanglement and supports canonical forms for controlled Hilbert-space truncation.
  • CLASSICAL SIMULATION TECHNIQUES: PEPS better matches higher-dimensional geometry but lacks canonical forms, making inference and time evolution vulnerable to truncation error.
  • CLASSICAL SIMULATION TECHNIQUES: PEPS can be efficient for sufficiently fast quenches, but expanding neighborhoods becomes prohibitive for long annealing times and higher dimensions.
  • CLASSICAL SIMULATION TECHNIQUES: NQS methods reproduce short-time dynamics but cannot achieve acceptable accuracy beyond the shortest times and smallest models examined.

TWO-DIMENSIONAL SYSTEMS

In two-dimensional spin glasses, QPU observables closely match MPS ground truths while QPU error remains roughly size-independent and the classical resources needed for comparable MPS accuracy grow rapidly. PEPS and NQS work for fast quenches but fail to sustain QPU-level accuracy for slower or larger systems.

  • TWO-DIMENSIONAL SYSTEMS: QPU observables closely agree with MPS ground truths across nearly three orders of magnitude of annealing time for a 6×6 spin glass.
  • TWO-DIMENSIONAL SYSTEMS: For ta=7 ns, QPU samples and MPS output at χ=64 show closely resembling correlation and classical-fidelity measures, with χQ≈64.
  • TWO-DIMENSIONAL SYSTEMS: QPU median correlation error remains roughly constant across system sizes, while matching it with MPS requires an increasing QPU-equivalent bond dimension χQ.
  • TWO-DIMENSIONAL SYSTEMS: At ta=20 ns, PEPS and NQS struggle even on relatively small systems, while NQS effort grows prohibitively with system size by ta=7 ns.
  • TWO-DIMENSIONAL SYSTEMS: PEPS performs well at ta=2 ns and improves systematically at ta=7 ns, but neighborhood-limited errors remain above QPU level at ta=20 ns.
  • TWO-DIMENSIONAL SYSTEMS: The study finds that PEPS and NQS do not match MPS utility as competitive solvers for higher-dimensional topologies.

HIGHER-DIMENSIONAL SYSTEMS

Higher-dimensional spin-glass topologies show area-law entanglement scaling, while QPU observables agree closely with converged classical ground truths at simulable sizes and support universal scaling beyond them.

  • Entanglement scaling: The required MPS bond dimension χQ grows exponentially with bipartition area across cubic, diamond, and biclique topologies.The relevant area scales to leading order as N^(d−1)/d, consistent with area-law entanglement scaling.
  • Entanglement scaling: Maximum entanglement entropy Smax varies linearly with log χQ across topologies, system sizes, and quench rates.At χQ, MPS entropy closely matches ground-truth entropy, indicating that QPU-quality MPS captures nearly all system entanglement.
  • Dynamical scaling: QPU and MPS ground truths differed by a median of only 1% for the largest converged lattices across topologies and annealing times.The comparison used the spin-glass order parameter ⟨q2⟩ for 20-instance ensembles at ta = 7 ns and 20 ns.
  • Dynamical scaling: Dynamic finite-size scaling collapses Binder cumulants across classically inaccessible sizes and extracts Kibble–Zurek exponents for the studied topologies.The collapsed sizes reach N = 567 for diamond systems, with the collapse showing a crossover from power-law to system-spanning correlations.
  • Resource estimates: Extrapolated MPS calculations would require millions of years per input on Frontier for the largest problems, with memory and energy demands exceeding stated practical capacities.The estimate assumes perfect parallelization and applies specifically to MPS, the only evaluated method matching QPU quality across all quench times.

SUMMARY AND CONCLUSIONS

The paper demonstrates beyond-classical computation for nonequilibrium magnetic spin dynamics across several programmable topologies. QPU results agree with quantum theory at small and large scales, while classical approximate methods face severe resource limitations.

  • SUMMARY AND CONCLUSIONS: The study demonstrates beyond-classical computation for nonequilibrium magnetic spin dynamics on square, cubic, diamond, and biclique topologies.These topologies are relevant to materials science and AI and support scaling analysis through entanglement area laws and universal critical scaling.
  • SUMMARY AND CONCLUSIONS: Two distinct QPU generations produced consistent solutions agreeing with quantum theory across classically verifiable and larger-scale regimes.Small systems were verified against Schrödinger-equation simulations; larger systems followed anticipated universal quantum critical dynamic scaling.
  • SUMMARY AND CONCLUSIONS: PEPS and NQS approaches failed on modestly sized 2D problems, whereas MPS was the only evaluated method with a reliable scaling matching QPU quality.The comparison targeted approximate solutions to the Schrödinger equation at QPU-level accuracy.
  • SUMMARY AND CONCLUSIONS: MPS required bond dimensions with stretched-exponential dependence on qubit number, enabling extrapolation to larger systems but implying infeasible classical resources.For a few hundred qubits, projected Frontier runtime surpassed millions of years, with infeasible memory and energy requirements.

DATA AVAILABILITY

The study provides code and data resources while documenting the processors, calibration procedures, model evolution, and entanglement-scaling assumptions underlying its simulations.

  • DATA AVAILABILITY: Code and data are available through the Zenodo repository, with MPS and PEPS experiments using ITensor and yastn.
  • Experimental platforms: The experiments used ADV1 and ADV2 superconducting annealing processors, with ADV2 selected for main measurements because of its lower noise and higher energy scale.
  • Calibration: Model quench times were mapped to nominal QPU times and an energy-scale factor κ using calibration on an ensemble of 20 2D spin glasses.
  • Entanglement scaling: Quenched dynamics show area-law entanglement, while collecting multiple qubits into one tensor cannot bypass the resulting scaling constraints.

3. Classical computational platform limitations

The paper evaluates classical tensor-network and related approaches under the entanglement, correlation, contraction, and hardware constraints encountered in higher-dimensional quantum quenches.

  • MPS: Current MPS-TDVP approaches become prohibitive for coherent anneal quenches with several hundred or thousand qubits at sufficiently large annealing times.
  • MPS: MPS efficiency depends critically on tensor ordering, which is generally difficult to optimize for generic higher-dimensional quantum states.
  • Tensor-network scaling: Tree tensor networks cannot provide a scalable advantage over MPS under area-law entanglement because their worst partition scales as N^(d−1)/d.
  • PEPS: PEPS is accurate at very short times, but longer times and larger energy scales increase correlation length and require larger neighborhoods, raising cost and systematic error.
  • Circuit resources: A 4×4 cylindrical lattice at 7 ns would require circuit depth 2800 and 39,200 two-qubit gates in a gate-model implementation.
  • Approximation stability: Locally tree-like approximations become unreliable as correlation length grows, with singular-value inversion potentially amplifying errors.

5. Non-canonical MPS in 1D

Non-canonical MPS simulations require sufficiently large clusters to capture correlations, while longer quenches demand larger bond dimensions and incur steeper computational costs.

  • Error estimation: The truncation-error estimate Δg is used as a rough observable-error proxy because it correlates with final expectation-value errors.The estimate becomes time-step independent as dt becomes small.
  • Cluster-size effects: Correlation error decreases with cluster size until saturating, while short clusters improve slowly with increasing bond dimension D.The minimal converged cluster size depends on annealing time.
  • Cluster-size effects: At ta = 7 ns, Mc ≈ 16, whereas at ta = 28 ns, Mc ≈ 64, consistent with Mc ∝ ta over the studied range.The paper relates this steeper scaling to quasiparticle motion and a second entanglement length scale.
  • Error estimation: The correlation error is roughly an order of magnitude lower than the corresponding total truncation error across tested environments and bond dimensions.This relation is observed for multiple annealing times and simulation settings.
  • Cluster-size effects: The 1D model shows that clusters must be comparable to the correlation range, with more stringent limitations expected in 2D and 3D.Insufficient cluster size prevents efficient use of the available bond dimension.

6. 2D PEPS

The 2D PEPS study expands local truncation environments and evaluates their accuracy-cost trade-offs. PEPS can match or surpass QPU accuracy in limited regimes, but systematic errors exceed QPU results for longer, harder quenches.

  • Methods: The baseline neighborhood tensor update contracts an eight-site cluster exactly at cost O(D^8 + pD^7).The cluster contains two central sites and six nearest neighbors.
  • Methods: Expanded neighborhood contractions preserve the same O(D^8 + pD^7) asymptotic cost while incorporating partial information from the next tensor layer.Further extensions add larger neighborhoods and loops.
  • Methods: Plaquette PEPS covers four times as many qubits per truncation cluster and can require less than the squared one-site-NTU bond dimension for comparable truncation error.The larger effective cluster improves use of the truncated bond dimension.
  • Error analysis: The correlation error is roughly Δ/10 across tested PEPS environments, bond dimensions, and annealing times.This supports truncation error as a practical quality metric for PEPS time evolution.
  • Results: For ta = 2 ns all tested methods are below QPU precision, whereas all tested methods are worse than QPU results for 20 ns quenches.A plaquette PEPS with NN+ environments is identified as a leading method for 7 ns quenches.
  • 3D extension: For cubic PEPS, increasing cluster size can reduce errors more effectively than increasing D, but sufficiently long anneals inevitably become D-limited.The reported scheme-D error crosses the 2% QPU benchmark around 4 ns for D = 4, 6.

IV. MEASURING THE QUALITY OF SAMPLED DISTRIBUTIONS

The paper evaluates sampled quantum-state distributions with correlation error and classical fidelity, while accounting for sampling bias and fidelity-based estimates of QPU-equivalent MPS bond dimension. These comparisons support the connection between correlation accuracy and broader distributional agreement.

  • The sampling task is approximately independent and identically distributed sampling from a quenched quantum state in the computational basis.
  • Correlation error and classical fidelity are used to measure the quality of sampled distributions.
  • Classical fidelity can reveal distribution differences beyond two-body correlations, including sign-symmetry breaking that correlation error misses.
  • When χ = ˜χ = 256, remaining distribution error is sampling error, while QPU samples appear roughly equivalent to MPS with χ = 64.
  • Fidelity-based QPU-equivalent bond dimensions deviate downward for larger, higher-entropy problems because estimating fidelity requires exponentially many samples, yet correlation- and fidelity-based estimates agree quantitatively before this effect dominates.
  • Partial MPS extrapolation improves errors by at most a factor of 2 in bond dimension and therefore does not change the competitiveness of approximate MPS methods.
  • MPS must retain nearly full entanglement during the quench to match QPU quality; stopping early raises correlation errors above QPU levels.

F. Area-law scaling for slower quenches

For slower quenches, correlation errors are slightly higher but show less size-dependent growth in biclique systems, while the required QPU-equivalent MPS bond dimension is lower. The reported entanglement and bond-dimension trends remain consistent with area-law scaling.

  • F. Area-law scaling for slower quenches: For ta = 20 ns, correlation errors are slightly higher across topologies, with less size-dependent growth for biclique problems than at 7 ns.
  • F. Area-law scaling for slower quenches: Longer quenches have lower χQ in all topologies, allowing larger lattices to be evaluated in some cases.
  • F. Area-law scaling for slower quenches: MPS correlation error is evaluated against dt → 0 for 8×8 square, 3×3×3 cubic, and K8,8 biclique systems at 7 ns and 20 ns.
  • F. Area-law scaling for slower quenches: In oblong cylinders, QPU correlation error is flat with N = 6Ly, while χQ is nearly flat for 7 ns and shows slight scaling for 20 ns.
  • F. Area-law scaling for slower quenches: Smax increases gently with Ly and remains similarly related to log(χQ) as in other ensembles.

VII. NEURAL NETWORKS

The paper tests neural quantum-state approaches for real-time TFIM dynamics alongside tensor-network methods. Their performance is limited by sampling, optimization, stability, expressiveness, and scaling constraints, especially beyond small square lattices.

  • NQS methods construct computationally efficient wave-function ansätze, using Markov-chain or direct sampling depending on the neural architecture.
  • tVMC updates variational parameters by estimating a force vector and the Fubini–Study metric matrix S at each time step.
  • Because S is p×p, inversion costs O(p^3), while stable simulations typically require Ns ∼10^4 samples per time step.
  • ptVMC minimizes infidelity between variational states at successive times and permits models with unrestricted parameter counts.
  • A Jastrow ansatz was stable but insufficiently expressive, feed-forward networks failed at larger sizes, and convolutional networks were ineffective for the nonsymmetric spin-glass Hamiltonian.
  • Autoregressive models fail for tVMC, while cRBM ptVMC is more stable but less accurate than tVMC, especially at larger system sizes.
  • NQS studies were limited to square lattices and found that larger or longer-time simulations could exhaust GPU resources; experiments at 20 ns were too costly.

VIII. CLASSICAL COMPUTE RESOURCES

The classical-resource analysis extrapolates measured MPS costs to estimate the computational burden of matching QPU quality. The resulting time, memory, and energy requirements become infeasible at the largest studied scales.

  • The study uses heterogeneous CPU and GPU resources, including Summit, Frontier, and several institutional clusters.
  • Resource extrapolation starts from a 54-qubit Summit simulation requiring 12 GB of memory and 7 × 10^4 seconds at χ = 724 and ta = 7 ns.
  • The extrapolation assumes a constant time step dt = 0.01 ns throughout.
  • Matching QPU quality on the largest systems would require more than millions of years using all of Frontier under the reported comparison.
  • The estimated MPS memory requirements exceed Frontier node flash memory of 4 TB and total filesystem storage of 700 PB.
  • The critical-exponent table compares QPU Binder-cumulant estimates with independent classical Monte Carlo and theoretical predictions across square, cubic, and biclique topologies.

IX. CRITICAL EXPONENTS AND DYNAMIC FINITE-SIZE SCALING ANSATZ

The paper estimates Kibble–Zurek exponents by collapsing Binder-cumulant data across system sizes and quench times, using universal critical scaling to validate QA dynamics beyond classically verifiable sizes.

  • Universality classes: The analysis covers square, cubic, cubic no-dimer, diamond, and biclique systems, assigned to 2D Ising, 3D Ising, or mean-field universality classes.For bicliques, the effective linear size is L = M^1/8 based on the upper-critical dimension 8.
  • Dynamic finite-size scaling: Binder-cumulant collapse uses the variable taL^-µ, with µ = z + 1/ν, to estimate dynamic scaling across topologies.U is preferred because it requires collapsing only µ, whereas ⟨q2⟩ also requires the anomalous dimension η.
  • Exponent estimates: µ = 2.6(3) and µ = 3.27(41) follow from the reported z and 1/ν estimates for two scaling analyses.These estimates are compared with classical Monte Carlo and mean-field predictions.
  • Exponent estimates: The measured low-precision-glass estimates, 2.67(30) and 2.99(19), agree with corresponding classical estimates within reported uncertainty.The paper notes that larger QPUs may clarify finite-size effects.
  • Exponent estimates: For bicliques, estimates from M = 8 to 24 are close to mean-field expectations despite large per-size fluctuations.The expected collapse rescales ta by M^-µ/8 ≈ M^-3/4.
  • Validation beyond classical reach: The agreement in critical scaling provides indirect validation of QA results beyond the classically simulable regime, complementing precise small-system comparisons.For 1D systems, the measured exponent is µ = 2.04(5), consistent with theory.

A. Cluster approximations with MPS

MPS-based cluster approximations estimate local correlations by simulating smaller displaced clusters, reducing numerical cost but losing accuracy as quench times and correlation lengths increase.

  • Cluster construction: MPS simulations on displaced open-boundary clusters provide local-correlation estimates for larger square-lattice systems.Cluster positions are varied so that all short-range correlations can be obtained.
  • Cluster construction: Three traveling-cluster geometries are tested: square clusters, open stripes, and periodic stripes.The corresponding sizes are Lc×Lc, Lc×Ly, and Lx×Lc.
  • Accuracy and cost: For ta = 0.3 ns, length-4 clusters on 6×6 systems achieve QPU-level accuracy, while length-5 clusters remain trustworthy up to ta = 2 ns.All three tested cluster shapes perform well in this short-time regime.
  • Accuracy and cost: Correlation error grows with ta because longer correlation lengths expose correlations inaccessible to small clusters.The paper uses this trend to connect cluster requirements with increasing dynamical correlation lengths.
  • Accuracy and cost: Cluster simulations can be parallelized, producing linear numerical-cost growth with system size for square and boundary-spanning stripe clusters.The relevant scaling is N for square clusters and L for boundary-spanning stripes.
  • Accuracy and cost: Stripe clusters cost more than equivalent-width square clusters because their greater length and higher required bond dimension accommodate more entanglement.The paper anticipates logarithmic entropy corrections near the quantum critical point.

1. Two dimensions

Rotating-cluster approximations decompose lattices into overlapping blocks and recombine their local correlations; they work for short-ranged 2D dynamics but fail increasingly for slower quenches and higher-dimensional systems.

  • Two dimensions: RCA chooses block width W, deletes periodic couplers at offsets, simulates all W×L blocks, and combines correlations using distance from block cuts.Long-distance correlations may be inferred from short-range correlations or set to zero.
  • Two dimensions: For L = 8 square lattices and ta < 4 ns, W = 4 RCA is indistinguishable from full-lattice QPU simulation because correlations are short-ranged.At longer ta, RCA error becomes significantly larger than full-lattice error.
  • Two dimensions: W ≈ 6 approaches the noise floor at ta = 7 ns, whereas W ≈ 9 is required at ta = 20 ns, consistent with increasing correlation length.The comparison uses µ ≈ 3 for 2D systems.
  • Two dimensions: For L = 18 and ta = 20 ns, W = 6 RCA error exceeds twice the L = 8 full-lattice error, indicating insufficient cluster width for QPU-level accuracy.This inference assumes exact simulation of the clusters.
  • Scope: The 2D analysis concerns correlation error and does not establish routine full-state sampling from cluster outputs.It supports only the possibility of spoofing spin-spin correlations for L = 18 and ta = 20 ns.
  • Higher dimensions: In 3D, W = 4 can spoof correlations up to ta = 7 ns, but significant deviations remain at ta = 20 ns even when W = 6.This pattern occurs for cubic dimer systems and is also observed for cubic no-dimer and diamond lattices.
  • Higher dimensions: Spatial decomposition is unpromising for cubic, cubic no-dimer, and diamond systems in this regime, consistent with significant system-spanning correlations.Such correlations limit related approximation methods.

C. Classical extrapolation of statistics from local to global

The paper tests whether global distributions can be reconstructed from local correlations using fully visible Boltzmann models, finding success when local estimates are accurate and correlations remain sufficiently tractable.

  • Classical extrapolation: Fully visible Boltzmann distributions trained on accurate local pairwise correlations can infer long-range correlations and approximately sample full states.The demonstrated accuracy can be comparable to the QPU when sampling a ground truth.
  • Model and assumptions: The Boltzmann model represents P(x) = 1/Z exp(θifi(x)), with parameters inferred from expectations of the chosen features fi(x).For geometrically local features such as nearby x_i x_j, local statistics can suffice when long-range correlations decay classically.
  • Model and assumptions: Training is practically straightforward at the presented lattice scales, although learning from sufficient statistics is technically NP-complete.Maximum pseudolikelihood offers an alternative efficient inference route.
  • Extrapolation results: At ta = 7 ns, ideal local data through geometric distance 3 reproduces the distribution with accuracy comparable to the ground truth.Rotating-cluster boundary effects leave εc just above the target threshold.
  • Extrapolation results: At ta = 20 ns, ideal data requires longer-distance correlations, while QPU[W=8] estimates become too poor for strong approximation.Increasing geometric distance improves ideal-data estimation but worsens the approximation-data model because of boundary errors.
  • Limitations: The method is favored by less-entangled distributions but degraded by increasing correlation lengths and dimensionality, which reduce the quality of local cluster estimates.Its assumptions include weak, classically describable long-range correlations and accurate local observables.
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