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arxiv: 2604.26534 · v2 · submitted 2026-04-29 · 🪐 quant-ph

Neural and Tensor Networks in the Study of Quantum Annealing Processors

Tomasz \'Smierzchalski This is my paper

Pith reviewed 2026-05-07 13:18 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum annealingD-Wave processorstensor networksPEPSthermodynamic costbenchmarkingIsing problemssampling
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The pith

Reliable quantum annealing benchmarks must jointly measure solution quality and thermodynamic costs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a benchmarking framework for D-Wave quantum annealers that goes beyond simple energy comparisons by incorporating classical baselines, sampling metrics, and physical costs. It introduces SpinGlassPEPS.jl, a tensor-network heuristic that maps Ising problems to Potts clusters on Pegasus-like graphs and uses approximate PEPS contractions for optimization and sampling. The work also models annealers as thermal machines, relating success rates to entropy production and dissipation, and shows that well-placed pauses can boost performance while lowering costs. Overall the thesis contends that trustworthy assessment of quantum annealing requires tracking both algorithmic output and physical expenditure.

Core claim

The central claim is that quantum annealing processors should be benchmarked as effective thermal machines whose success probability and solution quality are tied to measurable dissipation, entropy production, and effective temperature, with the SpinGlassPEPS.jl tensor-network solver supplying topology-aware classical references that make these thermodynamic relations testable.

What carries the argument

SpinGlassPEPS.jl, a topology-aware PEPS tensor-network solver that converts Ising instances to local Potts clusters, approximates the partition function, and performs branch-and-bound search in probability space, together with thermodynamic relations linking pauses, longitudinal fields, and entropy production to annealing performance.

If this is right

  • Strategic pauses in annealing schedules can simultaneously raise success probability and reduce dissipation and entropy production.
  • Longitudinal fields can become harmful when combined with paused schedules.
  • Reinforcement-learning post-processing can further improve the diversity and quality of samples returned by the annealer.
  • Exact small-system simulations can expose details of the underlying annealing dynamics that are otherwise hidden.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Energy-only comparisons may systematically overstate quantum-annealing advantage by omitting hidden physical costs.
  • The same joint performance-cost lens could be applied to other quantum optimization platforms beyond D-Wave.
  • Tensor-network heuristics of this type offer physically interpretable reference points that help diagnose when quantum devices are operating outside their modeled regime.

Load-bearing premise

The approximate PEPS contractions and thermodynamic relations derived from pauses and fields accurately capture the behavior of real D-Wave processors without significant unaccounted errors on large instances.

What would settle it

Direct experimental comparison on large instances showing that predicted thermodynamic costs or success probabilities from the PEPS model and pause analysis deviate substantially from measured values on actual D-Wave hardware.

Figures

Figures reproduced from arXiv: 2604.26534 by Tomasz \'Smierzchalski.

Figure 1.1
Figure 1.1. Figure 1.1: Symbolic representation of an example Ising spin glass system. It’s defined on a view at source ↗
Figure 1.2
Figure 1.2. Figure 1.2: Schematic representation of parallel annealing algorithm. The analog spin view at source ↗
Figure 1.3
Figure 1.3. Figure 1.3: Time to obtain ground state for each instance. Due to the small size of the view at source ↗
Figure 2.1
Figure 2.1. Figure 2.1: Example energy paths of adiabatic evolution. The expression view at source ↗
Figure 2.2
Figure 2.2. Figure 2.2: Example of schedule functions A(s) and B(s) for a representative D-Wave QPU (values shown as frequencies in the manufacturer’s convention). The shaded mid-anneal window indicates the region where the driver and problem energy scales compete most strongly (A(s) ≈ B(s)), often coinciding with an increased probability of diabatic excitations [16]. 2.3 D-Wave’s quantum annealers D-Wave quantum annealers are … view at source ↗
Figure 2.3
Figure 2.3. Figure 2.3: Connectivity structures of D-Wave Quantum Processing Units (QPUs), also view at source ↗
Figure 2.4
Figure 2.4. Figure 2.4: Example minor embedding of the complete graph view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Branch-and-bound search in probability space. The solution is built sequentially view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Execution flow of SpinGlassPEPS.jl. (a) An Ising instance on the native interaction graph is transformed into (b) a local Potts Hamiltonian on a king’s graph via clustering. (c) The resulting (Potts) partition function is represented as a PEPS tensor network on a square lattice. (d) Conditional marginals required by the search are computed by approximately contracting the PEPS. (e) A branch-and-bound swe… view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Transforming hardware graphs into Potts clusters. Black dots denote physical view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Edge-specific projections for compressing Potts pair energies. A naive repre view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Data flow between SpinGlassPEPS.jl sub-packages. An input Ising instance is parsed by SpinGlassNetworks.jl, which constructs the interaction graph, performs cluster￾ing, and outputs a Potts Hamiltonian. The Potts instance and solver parameters (e.g., β, search width M, and contraction settings such as maximal bond dimension χ) are passed to SpinGlassEngine.jl, which executes the branch-and-bound search i… view at source ↗
Figure 3.6
Figure 3.6. Figure 3.6: Example of lattice graphs represented by view at source ↗
Figure 3.7
Figure 3.7. Figure 3.7: Construction of PEPS tensor network from clustered Hamiltonian (Potts model) view at source ↗
Figure 3.8
Figure 3.8. Figure 3.8: Site tensor. Four virtual legs mediate the interaction structure and trace out view at source ↗
Figure 3.9
Figure 3.9. Figure 3.9: Additional structures. A diagonal tensor (a) carries diagonal interactions. Square view at source ↗
Figure 3.10
Figure 3.10. Figure 3.10: Computation of conditional probabilities via approximate PEPS contraction. view at source ↗
Figure 3.11
Figure 3.11. Figure 3.11: Triple-junction inpainting benchmark [117, 119]. Panel (a) shows the input: pixel labels are fixed on a circular boundary, while the interior (black region) is unknown. Panel (b) shows a lowest-energy configuration found by SpinGlassPEPS.jl. Due to the grid￾based discretization, the energy function is mildly anisotropic, which can bias reconstructed interfaces toward axis-parallel directions [117]. Pane… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Time-to-approximation-ratio on Pegasus-native Ising instances. Each panel shows view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Time-to-approximation-ratio on Zephyr-native Ising instances. Each panel shows view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Time-to-approximation-ratio on lattice instances. Each panel shows the ap view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Time-to-diversity-ratio for native Pegasus instances. We set view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Time-to-diversity-ratio for native Zephyr instances. We set view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Schematic representation of a D-Wave processor operating under a reverse view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Ground-state success probability PGS and average solution quality QGS for an antiferromagnetic Ising chain with N = 300 spins, obtained using reverse annealing schedules with and without a mid-anneal pause and for different values of the external magnetic field h. Each data point is averaged over 100 annealing runs with 10 samples per run. Interestingly, when an external magnetic field is present, the in… view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Thermodynamic and computational efficiency for an antiferromagnetic Ising view at source ↗
Figure 5.4
Figure 5.4. Figure 5.4: Thermodynamic response of a one-dimensional Ising chain ( view at source ↗
Figure 5.5
Figure 5.5. Figure 5.5: Two-dimensional uniform instance on Pegasus. Scan of the reverse￾annealing turning point s for cycle times τ = 100, 200, 500, 1000 µs. Panels show (a) pseudo￾likelihood effective bath temperature T2(s) (with T1 as reference), (b) TUR-certified lower bound on the average work per spin ⟨W⟩/L, (c) corresponding power bound ⟨P⟩/L, (d) mean processor energy change ⟨∆E1⟩/L, (e) TUR lower bound on dumped heat −… view at source ↗
Figure 5.6
Figure 5.6. Figure 5.6: Two-dimensional constant instance on Pegasus. Same analysis as view at source ↗
Figure 5.7
Figure 5.7. Figure 5.7: Two-dimensional CBFM instance on Pegasus. Same analysis as view at source ↗
Figure 5.8
Figure 5.8. Figure 5.8: Operating-mode phase diagrams of Pegasus and Zephyr. Operating regimes in the (β1, s) plane for (a) Pegasus and (b) Zephyr. Colors denote engine (E), refrigerator (R), heater (H), and accelerator (A) modes. Color intensity indicates the ideal Carnot efficiency or coefficient of performance for orientation. Finite-time operation remains below these ideal limits. Engine operation appears only when the prep… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: Overview of our method. Arrows represent consecutive steps. First, we define view at source ↗
Figure 6.2
Figure 6.2. Figure 6.2: Performance of simulated annealing with reinforcement (SAwR) compared to view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: Example of a natural embedding of complete graph view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Median total variation distance (left) and classical fidelity view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Median ground-state probability as a function of the fast-anneal duration view at source ↗
read the original abstract

Quantum annealing targets low-energy solutions of Ising/QUBO problems, but reliable assessment requires more than best-energy comparisons. This dissertation develops a benchmarking framework for D-Wave quantum annealers that combines strong classical baselines, sampling and diversity metrics, and thermodynamic cost. Its first contribution, SpinGlassPEPS$.$jl, is a topology-aware tensor-network heuristic for optimization and sampling on Pegasus/Zephyr-like graphs. It maps Ising instances to local Potts clusters, represents the partition function with PEPS, and performs branch-and-bound search in probability space. Benchmarks show that it is a physically interpretable reference solver, though approximate contractions limit its competitiveness on the largest instances. The second contribution treats quantum annealers as effective thermal machines, relating success probability and solution quality to dissipation, entropy production, and effective temperature. Carefully placed pauses can improve performance while reducing thermodynamic cost, although longitudinal fields may become harmful in paused schedules. The thesis also introduces reinforcement-learning post-processing to improve returned samples and exact small-system simulations to probe annealing dynamics. Overall, it argues for quantum-annealing benchmarks that jointly measure algorithmic performance and physical expenditure.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The dissertation develops a benchmarking framework for D-Wave quantum annealers that integrates a topology-aware tensor-network solver (SpinGlassPEPS.jl) for optimization and sampling on Pegasus/Zephyr graphs, treats the annealer as an effective thermal machine to relate success probability and solution quality to dissipation/entropy production/effective temperature, applies reinforcement-learning post-processing to samples, and uses exact small-system simulations. It argues that reliable assessment requires joint measurement of algorithmic performance and physical expenditure, with results indicating that carefully placed pauses can improve performance while lowering thermodynamic cost (though longitudinal fields may become harmful in paused schedules).

Significance. If the PEPS approximations and thermodynamic mappings are shown to be sufficiently accurate, the framework would provide a physically grounded way to compare annealing schedules on both success metrics and energy costs, potentially guiding more efficient processor operation. The SpinGlassPEPS.jl contribution supplies a new classical reference tool for these graphs, and the overall emphasis on joint performance-cost benchmarks addresses a gap in current QA evaluation practices.

major comments (3)
  1. [thermodynamic analysis of pauses and fields] Thermodynamic cost analysis: the relations linking success probability to dissipation and entropy production via pauses and fields depend on effective-temperature derivations; the manuscript does not state or demonstrate that these temperatures are derived independently of the success-probability data used to validate the joint benchmark, which is load-bearing for the central claim that physical expenditure can be meaningfully compared to algorithmic metrics.
  2. [SpinGlassPEPS.jl tensor-network heuristic] SpinGlassPEPS.jl contribution: approximate PEPS contractions are noted to limit competitiveness on the largest instances, yet no quantitative error bounds, truncation-error estimates, or validation against exact partition functions are supplied for Pegasus/Zephyr graphs at D-Wave scales; without these, the solver cannot serve as a trustworthy classical reference for the proposed joint benchmarks.
  3. [pause and field schedule experiments] Experimental claims on pauses and longitudinal fields: the statements that pauses improve performance while reducing thermodynamic cost and that fields become harmful rest on unspecified experiments; details on instance selection, error bars, and comparison baselines are absent, preventing assessment of whether the observed effects support the broader benchmarking recommendation.
minor comments (2)
  1. [abstract and introduction] The abstract and introduction could more explicitly separate the contributions (tensor-network solver, thermal-machine model, RL post-processing) and state the precise scope of the D-Wave instances studied.
  2. [thermodynamic sections] Notation for effective temperature and entropy production should be defined once and used consistently across the thermodynamic sections to avoid ambiguity when comparing schedules.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas where the manuscript can be clarified and strengthened. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [thermodynamic analysis of pauses and fields] Thermodynamic cost analysis: the relations linking success probability to dissipation and entropy production via pauses and fields depend on effective-temperature derivations; the manuscript does not state or demonstrate that these temperatures are derived independently of the success-probability data used to validate the joint benchmark, which is load-bearing for the central claim that physical expenditure can be meaningfully compared to algorithmic metrics.

    Authors: The referee is correct that the manuscript does not explicitly demonstrate the independence of the effective-temperature derivation from the success-probability data. The effective temperatures are computed from the annealing schedule parameters and device calibration data using the standard effective thermal model, prior to and separately from the benchmark validation. To address this, we will add a new subsection in the thermodynamic analysis chapter that details the derivation procedure, references the calibration experiments, and shows that the temperature values are fixed before evaluating success probabilities. This will make the separation explicit and support the joint benchmark claim. revision: yes

  2. Referee: [SpinGlassPEPS.jl tensor-network heuristic] SpinGlassPEPS.jl contribution: approximate PEPS contractions are noted to limit competitiveness on the largest instances, yet no quantitative error bounds, truncation-error estimates, or validation against exact partition functions are supplied for Pegasus/Zephyr graphs at D-Wave scales; without these, the solver cannot serve as a trustworthy classical reference for the proposed joint benchmarks.

    Authors: We agree that the current manuscript lacks quantitative error analysis for the PEPS approximations on Pegasus and Zephyr graphs. Although the text notes the limitations of approximate contractions, it does not supply truncation-error estimates or direct validations against exact partition functions. In the revised version we will include bond-dimension scaling studies, error bounds derived from truncation thresholds, and comparisons to exact results on small instances of these graphs. For larger scales we will add a discussion of error propagation. These additions will better qualify SpinGlassPEPS.jl as a reference solver. revision: yes

  3. Referee: [pause and field schedule experiments] Experimental claims on pauses and longitudinal fields: the statements that pauses improve performance while reducing thermodynamic cost and that fields become harmful rest on unspecified experiments; details on instance selection, error bars, and comparison baselines are absent, preventing assessment of whether the observed effects support the broader benchmarking recommendation.

    Authors: The referee correctly notes that experimental details are insufficiently specified. The reported effects are based on experiments using randomly generated Ising instances with Pegasus topology, executed with multiple independent runs on the D-Wave device to obtain statistical error bars, and compared against standard forward-annealing baselines without pauses. We will expand the experimental section to include the precise instance-generation protocol, the number of instances and repetitions, the statistical methods for error bars, and direct side-by-side comparisons to non-paused schedules. This will allow readers to evaluate the support for the claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper outlines three main independent contributions: the SpinGlassPEPS.jl tensor-network solver as a classical reference for Pegasus/Zephyr graphs, a thermodynamic mapping of quantum annealers to effective thermal machines that relates success probability and solution quality to dissipation and entropy production via pauses and fields, and auxiliary RL post-processing plus exact small-system simulations. No load-bearing steps reduce by construction to fitted inputs or self-citations; the thermodynamic relations are presented as derived from observable pauses and fields rather than tautological redefinitions of success probability, and the PEPS contractions serve as an external heuristic benchmark without the paper claiming they are forced by the target D-Wave data. The framework therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review yields no explicit free parameters, axioms, or invented entities; the central claims implicitly rest on unstated assumptions about the accuracy of PEPS approximations and the validity of mapping annealing dynamics to thermodynamic quantities.

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