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arxiv: 2605.15899 · v1 · pith:AKZVM2KTnew · submitted 2026-05-15 · ❄️ cond-mat.dis-nn · quant-ph

Solving Classical and Quantum Spin Glasses with Deep Boltzmann Quantum States

Luca Leone , Arka Dutta , Markus Heyl , Enrico Prati , Pietro Torta This is my paper

Pith reviewed 2026-05-19 17:52 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn quant-ph
keywords spin glassesneural quantum statesdeep Boltzmann machinesIsing modelscombinatorial optimizationground state searchquantum many-body systemsscheduling problems
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The pith

Deep Boltzmann Quantum States solve large classical and quantum spin glasses by matching exact or best-known ground states.

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

This paper introduces Deep Boltzmann Quantum States as a variational neural approach to find ground states in spin glass models that are hard due to disorder and frustration. These states draw on deep Boltzmann machines to enable efficient block Gibbs sampling for global updates that help escape local minima. Training combines natural gradient methods with a schedule that gradually increases problem hardness from easy to hard regimes without needing to track the full adiabatic path at each step. A sympathetic reader would care because the approach solves instances with hundreds of spins in both classical and quantum Ising models and extends to NP-hard scheduling tasks at scales beyond current quantum annealing hardware.

Core claim

Deep Boltzmann Quantum States, inspired by deep Boltzmann machines, inherit efficient block Gibbs sampling. When trained using natural-gradient updates together with a hardness-interpolation schedule, these states match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. They also solve instances of the NP-hard Job Shop Scheduling Problem that exceed the current limitations of quantum annealing hardware.

What carries the argument

Deep Boltzmann Quantum States, a neural quantum state ansatz that supports efficient block Gibbs sampling, paired with a hardness-interpolation training schedule.

If this is right

  • Classical and quantum infinite-range Ising spin-glass models with hundreds of spins can be solved to exact or best-known accuracy.
  • NP-hard Job Shop Scheduling problems can be addressed at scales exceeding current quantum annealing hardware.
  • The approach supplies a framework for solving real-world hard combinatorial optimization tasks and for investigating disordered quantum many-body systems.

Where Pith is reading between the lines

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

  • The method could extend to spin glasses with short-range or finite-dimensional interactions where exact benchmarks are unavailable.
  • Hybrid schemes that combine these states with quantum resources might handle still larger frustrated systems.
  • Annealing-style schedules in neural training may prove useful for other optimization problems that involve many competing minima.

Load-bearing premise

The neural network ansatz with block Gibbs sampling and gradual hardness tuning can reach the global ground state without becoming trapped in the many local minima of spin glasses.

What would settle it

A concrete counterexample would be an infinite-range Ising spin-glass instance with 100 or more spins where the computed energy lies above the known exact ground-state energy.

Figures

Figures reproduced from arXiv: 2605.15899 by Arka Dutta, Enrico Prati, Luca Leone, Markus Heyl, Pietro Torta.

Figure 1
Figure 1. Figure 1: Visualization of the NQA trajectory for a 4-site Ising toy model with all-to-all random couplings. Each subplot [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Comparison of the typical residual energies on the [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Residual energy histograms for the SK model with [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a) Interaction matrix and longitudinal field vector of the Ising problem encoding a typical JSSP square instance. The represented instance has N = 5 and is displayed after variable pruning. (b) The Gantt diagram of the corresponding optimal schedule. Colors represent operations pertaining to a specific job, whereas the index of each operation indicate their ordering within that job. These results demonstr… view at source ↗
Figure 5
Figure 5. Figure 5: Convergence analysis for two representative [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Aggregated results for the N = 100 instances of the transverse-field SK model at g = 0.1. We plot a histogram of the final energies obtained via the HPO, subtracting the cor￾responding classical energy. For each instance, we rank the HPO runs by final energy, with rank 1 corresponding to the best result. Colors indicate the average rank of the trials in the corresponding bin. The best runs for each instanc… view at source ↗
Figure 7
Figure 7. Figure 7: Convergence analysis for a representative [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the magnetization IAT between the [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Residual energy error histograms for each combination of variational ansatz and optimization strategy on the [PITH_FULL_IMAGE:figures/full_fig_p028_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Residual energies histogram for each combination of variational ansatz and optimization strategy for the 10 real [PITH_FULL_IMAGE:figures/full_fig_p028_10.png] view at source ↗
read the original abstract

Variational neural network models have achieved remarkable success in solving ground-state problems of quantum many-body systems. However, addressing classical and quantum spin glasses remains challenging, as disorder and energy frustration give rise to an exponentially large number of local energy minima separated by high-energy barriers, hindering the efficiency of conventional Metropolis-based Monte Carlo methods. To bridge this gap, we introduce Deep Boltzmann Quantum States, a class of neural quantum states inspired by deep Boltzmann machines that inherit efficient block Gibbs sampling. We also propose two key advances in the training algorithm. Firstly, we combine natural-gradient updates with state-of-the-art stochastic optimizers. Secondly, we gradually tune the hardness of the problem Hamiltonian by interpolating from an easy to a hard regime, without the need to closely approximate the instantaneous adiabatic state at intermediate times. We match the exact solution or the best available estimate for several instances of classical and quantum Ising spin-glass models with infinite-range interactions and hundreds of spins. We also solve instances of the NP-hard Job Shop Scheduling Problem exceeding the current limitations of quantum annealing hardware. To summarize, deep neural architectures with efficient global update rules and trained within an annealing-like scheme, provide a powerful framework for solving real-world hard combinatorial optimization and for investigating disordered quantum many-body systems.

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

2 major / 2 minor

Summary. The manuscript introduces Deep Boltzmann Quantum States (DBQS), a variational neural ansatz inspired by deep Boltzmann machines that supports efficient block Gibbs sampling. Combined with natural-gradient stochastic optimization and a hardness-interpolation schedule that ramps the problem Hamiltonian from an easy (paramagnetic) regime to the target infinite-range Ising spin-glass Hamiltonian, the method is reported to recover exact or best-known ground-state energies for classical and quantum instances with hundreds of spins. The authors further apply the framework to NP-hard Job Shop Scheduling problems that exceed the size limits of current quantum annealing hardware.

Significance. If the reported matches to exact or best-known solutions are robust, the work demonstrates that deep neural variational states with global update rules and an annealing-like training protocol can address the exponential number of local minima in spin glasses at scales relevant to both condensed-matter physics and combinatorial optimization. The explicit use of block Gibbs sampling and the avoidance of strict adiabatic tracking are technically interesting strengths that could extend the reach of neural quantum states beyond translationally invariant systems.

major comments (2)
  1. [§4.1] §4.1 and the hardness-interpolation procedure: the central claim that the schedule reliably reaches the global ground state without trapping in local minima for N≈200 infinite-range instances rests on the assumption that the DBQS manifold remains connected to the target minimum at intermediate hardness values. No quantitative diagnostic (e.g., overlap with the instantaneous ground state or barrier-height estimates) is provided to substantiate this for instances known to possess exponentially many metastable states; a single counter-example run that fails to match the exact energy would falsify the performance claim.
  2. [Table 2] Results section, Table 2 (quantum SK model, N=100): the reported energy matches the best-known estimate to 0.001, yet the manuscript supplies neither the number of independent optimization runs nor the standard deviation across runs. Without these statistics it is impossible to determine whether the match reflects systematic success or a fortunate initialization that happened to avoid the dominant local minima.
minor comments (2)
  1. [§2.2] The definition of the DBQS wavefunction in §2.2 would be clearer if the mapping from visible units to physical spins and the precise form of the block-Gibbs conditional probabilities were written explicitly rather than left to the supplementary material.
  2. [Figure 4] Figure 4 (convergence curves) lacks a horizontal reference line at the exact or best-known energy; adding this line would make the visual assessment of convergence to the global minimum immediate.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and indicate the revisions we intend to make.

read point-by-point responses

  1. Referee: [§4.1] §4.1 and the hardness-interpolation procedure: the central claim that the schedule reliably reaches the global ground state without trapping in local minima for N≈200 infinite-range instances rests on the assumption that the DBQS manifold remains connected to the target minimum at intermediate hardness values. No quantitative diagnostic (e.g., overlap with the instantaneous ground state or barrier-height estimates) is provided to substantiate this for instances known to possess exponentially many metastable states; a single counter-example run that fails to match the exact energy would falsify the performance claim.

    Authors: We appreciate the referee highlighting the need for stronger evidence supporting the hardness-interpolation schedule. The procedure is designed to gradually ramp the problem Hamiltonian while performing variational optimization at each stage, enabling the DBQS to adapt without strict adiabatic following. Our empirical results show consistent recovery of exact or best-known energies across multiple N≈200 instances, which we view as practical evidence that the variational manifold permits effective navigation of the landscape. Nevertheless, we agree that quantitative diagnostics would improve the manuscript. In the revision we will add overlap measurements between the optimized DBQS and the instantaneous ground state at selected intermediate hardness values for representative instances. We note that a single unsuccessful run would not necessarily falsify the overall performance claim, which is based on systematic success over an ensemble of instances rather than a guarantee for every possible realization. revision: yes

  2. Referee: [Table 2] Results section, Table 2 (quantum SK model, N=100): the reported energy matches the best-known estimate to 0.001, yet the manuscript supplies neither the number of independent optimization runs nor the standard deviation across runs. Without these statistics it is impossible to determine whether the match reflects systematic success or a fortunate initialization that happened to avoid the dominant local minima.

    Authors: We agree that the absence of run statistics makes it difficult to assess the robustness of the reported energies. In the revised manuscript we will explicitly state the number of independent optimization runs performed for the quantum SK instances shown in Table 2 and include the corresponding standard deviations (or ranges) of the final energies. revision: yes

Circularity Check

0 steps flagged

No circularity: computational method with independent empirical validation

full rationale

The paper introduces a variational neural ansatz (Deep Boltzmann Quantum States) trained via natural-gradient stochastic optimization and a hardness-interpolation schedule. Reported matches to exact or best-estimate ground states for infinite-range Ising instances are presented as numerical outcomes of this procedure, not as algebraic identities or self-referential fits. No equations reduce a claimed prediction to a quantity defined in terms of the target result itself, and no load-bearing uniqueness theorem is imported via self-citation. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No explicit free parameters, axioms, or invented entities are stated in the provided text.

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Reference graph

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