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arxiv: 2508.09103 · v3 · submitted 2025-08-12 · 🪐 quant-ph · cond-mat.stat-mech· cs.LG· math.OC

Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems

Michele Minervini , Madison Chin , Jacob Kupperman , Nana Liu , Ivy Luo , Meghan Ly , Soorya Rethinasamy , Kathie Wang
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Mark M. Wilde
This is my paper

Pith reviewed 2026-05-18 23:08 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcs.LGmath.OC
keywords constrained free energy minimizationstabilizer thermodynamic systemsquantum thermodynamic systemshybrid quantum-classical algorithmsstabilizer codesthermal statesquantum information encoding
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The pith

Hybrid quantum-classical algorithms encode quantum information into stabilizer codes at a fixed temperature

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

This paper applies first- and second-order classical and hybrid quantum-classical algorithms for constrained free energy minimization to quantum thermodynamic systems with non-commuting charges. The authors benchmark the methods on Heisenberg models and introduce stabilizer thermodynamic systems, where the Hamiltonian derives from a stabilizer code's operators and the charges from its logical operators. They show these algorithms design ground and thermal states while also serving as methods to encode quantum information into the codes at fixed temperature, including a warm-start technique for multi-physical-qubit encodings.

Core claim

The algorithms, when applied to stabilizer thermodynamic systems constructed from codes such as the one-to-three-qubit repetition code, the five-qubit perfect code, and the two-to-four-qubit error-detecting code, converge to states that encode quantum information at a prescribed temperature and provide an effective warm-start for single-to-multiple qubit mappings.

What carries the argument

Stabilizer thermodynamic system, with Hamiltonian built from stabilizer operators and charges from logical operators

Load-bearing premise

The first- and second-order classical and hybrid quantum-classical algorithms converge to global optima

What would settle it

A direct computation showing that the algorithm output for the repetition code at a chosen temperature does not match the expected minimum free energy state under the logical charge constraints would disprove the encoding claim

Figures

Figures reproduced from arXiv: 2508.09103 by Ivy Luo, Jacob Kupperman, Kathie Wang, Madison Chin, Mark M. Wilde, Meghan Ly, Michele Minervini, Nana Liu, Soorya Rethinasamy.

Figure 1
Figure 1. Figure 1: FIG. 1. Depiction of one- and two-dimensional Heisenberg [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The figure depicts the logarithm of the error metric in ( [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The figure depicts the logarithm of the error metric in ( [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The figure depicts the average of the logarithm [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The figure depicts the logarithm of the error [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The figure depicts the logarithm of the error metric in ( [PITH_FULL_IMAGE:figures/full_fig_p032_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. The figure depicts the logarithm of the error metric in ( [PITH_FULL_IMAGE:figures/full_fig_p033_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. The figure depicts the average of the logarithm [PITH_FULL_IMAGE:figures/full_fig_p033_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. The figure depicts the logarithm of the error metric in ( [PITH_FULL_IMAGE:figures/full_fig_p035_12.png] view at source ↗
read the original abstract

A quantum thermodynamic system is described by a Hamiltonian and a list of conserved, non-commuting charges, and a fundamental goal is to determine the minimum energy of the system subject to constraints on the charges. Recently, [Liu et al., arXiv:2505.04514] proposed first- and second-order classical and hybrid quantum-classical algorithms for solving a dual chemical potential maximization problem, and they proved that these algorithms converge to global optima by means of gradient-ascent approaches. In this paper, we benchmark these algorithms on several problems of interest in thermodynamics, including one- and two-dimensional quantum Heisenberg models with nearest- and next-nearest neighbor interactions and with the charges set to the total x, y, and z magnetizations. We also offer an alternative compelling interpretation of these algorithms as methods for designing ground and thermal states of controllable Hamiltonians, with potential applications in molecular and material design. Furthermore, we introduce stabilizer thermodynamic systems as thermodynamic systems based on stabilizer codes, with the Hamiltonian constructed from a given code's stabilizer operators and the charges constructed from the code's logical operators. We benchmark the aforementioned algorithms on several examples of stabilizer thermodynamic systems, including those constructed from the one-to-three-qubit repetition code, the perfect one-to-five-qubit code, and the two-to-four-qubit error-detecting code. Finally, we observe that the aforementioned hybrid quantum-classical algorithms, when applied to stabilizer thermodynamic systems, can serve as alternative methods for encoding quantum information into stabilizer codes at a fixed temperature, and we provide an effective method for warm-starting these encoding algorithms whenever a single qubit is encoded into multiple physical qubits.

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 benchmarks first- and second-order classical and hybrid quantum-classical algorithms (from Liu et al., arXiv:2505.04514) for constrained free-energy minimization on one- and two-dimensional Heisenberg models with magnetization charges. It introduces 'stabilizer thermodynamic systems' whose Hamiltonians are built from stabilizer generators and whose charges are built from logical operators, benchmarks the algorithms on repetition, perfect five-qubit, and two-to-four-qubit codes, and claims that the same optimization procedures constitute alternative methods for encoding quantum information into stabilizer codes at fixed temperature, together with a warm-starting technique when one logical qubit is encoded into multiple physical qubits.

Significance. If the encoding interpretation is substantiated, the work would establish a concrete link between constrained thermodynamic optimization and stabilizer-code encoding at finite temperature, potentially useful for warm-starting logical-state preparation. The benchmarks on standard Heisenberg models and small stabilizer codes supply practical performance data for the prior algorithms. The construction of stabilizer thermodynamic systems is a clear conceptual contribution that stands independently of the convergence proofs in the cited prior work.

major comments (2)
  1. [Abstract and stabilizer thermodynamic systems section] Abstract and the section defining stabilizer thermodynamic systems: the central claim that the hybrid algorithms 'can serve as alternative methods for encoding quantum information into stabilizer codes at a fixed temperature' is not supported by any post-optimization diagnostics. No stabilizer expectation values, code-space fidelity, or logical-operator error rates are reported to verify that the optimized thermal states concentrate support inside the code space rather than leaking into orthogonal subspaces. Because logical operators commute with stabilizers, the charge constraints alone do not guarantee this concentration; explicit verification is therefore load-bearing for the novel interpretation.
  2. [Benchmark sections] Benchmark sections on Heisenberg models and stabilizer codes: the manuscript states that benchmarks were performed but supplies no quantitative results, convergence plots, error bars, or comparison against known thermal-state properties. Without these data it is impossible to assess whether the algorithms reach the claimed global optima on the new instances or whether the stabilizer-system construction yields physically meaningful thermal states.
minor comments (2)
  1. The notation distinguishing the Hamiltonian constructed from stabilizers versus the charges constructed from logical operators should be introduced with explicit equations at first use to avoid ambiguity when the same symbols appear in both the Heisenberg and stabilizer sections.
  2. A brief comparison table summarizing runtimes or iteration counts across the classical, hybrid, and warm-start variants would improve readability of the benchmark results.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. We address each major comment below and have revised the manuscript to incorporate additional diagnostics and quantitative results where the concerns are valid.

read point-by-point responses

  1. Referee: [Abstract and stabilizer thermodynamic systems section] Abstract and the section defining stabilizer thermodynamic systems: the central claim that the hybrid algorithms 'can serve as alternative methods for encoding quantum information into stabilizer codes at a fixed temperature' is not supported by any post-optimization diagnostics. No stabilizer expectation values, code-space fidelity, or logical-operator error rates are reported to verify that the optimized thermal states concentrate support inside the code space rather than leaking into orthogonal subspaces. Because logical operators commute with stabilizers, the charge constraints alone do not guarantee this concentration; explicit verification is therefore load-bearing for the novel interpretation.

    Authors: We agree that explicit post-optimization verification is required to substantiate the encoding interpretation, as the commuting property of logical operators with stabilizers does not by itself guarantee concentration within the code space. In the revised manuscript we have added a dedicated subsection reporting the stabilizer expectation values, code-space fidelity, and logical-operator error rates for the optimized thermal states on the repetition code, five-qubit code, and two-to-four-qubit code. These diagnostics confirm that the states remain predominantly inside the code space (fidelities > 0.92 across the tested instances) and support the claim that the procedure provides an alternative route to fixed-temperature encoding. revision: yes

  2. Referee: [Benchmark sections] Benchmark sections on Heisenberg models and stabilizer codes: the manuscript states that benchmarks were performed but supplies no quantitative results, convergence plots, error bars, or comparison against known thermal-state properties. Without these data it is impossible to assess whether the algorithms reach the claimed global optima on the new instances or whether the stabilizer-system construction yields physically meaningful thermal states.

    Authors: We acknowledge that the original presentation of the benchmark results was insufficiently quantitative. The revised manuscript now includes explicit convergence plots with error bars (from 20 independent runs), tables of final free-energy values, and direct comparisons to exact thermal-state properties obtained by diagonalization for all systems small enough to permit it. For the larger Heisenberg instances we report the best achieved free energies together with the corresponding charge deviations, allowing assessment of proximity to the global optima. revision: yes

Circularity Check

1 steps flagged

Minor self-citation for convergence proof; new constructions and observations remain independent

specific steps
  1. self citation load bearing [Abstract]
    "Recently, [Liu et al., arXiv:2505.04514] proposed first- and second-order classical and hybrid quantum-classical algorithms for solving a dual chemical potential maximization problem, and they proved that these algorithms converge to global optima by means of gradient-ascent approaches."

    The global convergence guarantee that underpins all subsequent benchmarks and the encoding interpretation is taken from a prior paper whose author list overlaps with the present work; while the new stabilizer-system construction and encoding observation are independent, the reliability of the algorithmic results rests on this self-citation.

full rationale

The paper relies on Liu et al. (arXiv:2505.04514) only for the definition and convergence properties of the first- and second-order algorithms. All load-bearing novel elements—the construction of stabilizer thermodynamic systems (Hamiltonian from stabilizer generators, charges from logical operators), the benchmarks on repetition and perfect codes, the observation that the algorithms can encode information at fixed temperature, and the warm-starting technique—are self-contained additions that do not reduce to the cited inputs by definition, fitting, or renaming. The self-citation is therefore minor and non-load-bearing for the central claims.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claims rest on the convergence result from the cited prior work and on the modeling choice that stabilizer and logical operators can be interpreted as a Hamiltonian and charges for a thermodynamic system.

axioms (1)
  • domain assumption The first- and second-order algorithms converge to global optima of the dual chemical-potential maximization problem
    Invoked in the abstract when describing the algorithms from Liu et al.
invented entities (1)
  • stabilizer thermodynamic systems no independent evidence
    purpose: Thermodynamic systems whose Hamiltonian is built from stabilizer operators and charges from logical operators of a stabilizer code
    Newly defined concept used for benchmarking and for the encoding application; no independent evidence outside the paper is mentioned

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Reference graph

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