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arxiv: 2510.23546 · v2 · submitted 2025-10-27 · 🪐 quant-ph

Variational Thermal State Preparation on Digital Quantum Processors Assisted by Matrix Product States

Rui-Hao Li , Semeon Valgushev , Khadijeh Najafi This is my paper

Pith reviewed 2026-05-18 03:59 UTC · model grok-4.3

classification 🪐 quant-ph
keywords variational quantum algorithmsGibbs statesthermal state preparationmatrix product statesquantum simulationerror mitigationtransverse-field Ising model
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The pith

A variational framework uses classical matrix product states to guide quantum circuits toward accurate finite-temperature Gibbs states.

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

The paper introduces a hybrid variational method that evaluates the Helmholtz free energy classically with matrix product states and uses the result to optimize a hardware-efficient quantum ansatz for approximating thermal states. Benchmarking on one-dimensional chains up to 30 sites and two-dimensional lattices up to 6 by 6 shows observables such as energy density, susceptibility, and correlations matching exact or quantum Monte Carlo references across temperatures. The same approach is executed on a 156-qubit IBM Heron processor for a 30-site transverse-field Ising model, where error mitigation cuts relative errors in energy and susceptibility by more than half. A reader would care because reliable preparation of Gibbs states at finite temperature underpins quantum simulation of many-body systems and thermal sampling methods.

Core claim

By combining scalable classical matrix product state computation of the free energy with variational optimization of a hardware-efficient ansatz on quantum hardware, the method produces high-fidelity approximations to Gibbs states for one-dimensional lattices of 30 sites and two-dimensional lattices of 6 by 6 sites, with a working demonstration on a 156-qubit superconducting processor.

What carries the argument

Classical matrix product state evaluation of the Helmholtz free energy that guides variational optimization of a hardware-efficient quantum ansatz.

If this is right

  • Observables including energy density, susceptibility, specific heat, and two-point correlations match exact analytic results in one dimension and quantum Monte Carlo results in two dimensions.
  • The approach scales to systems requiring up to 44 qubits in numerical simulations.
  • Error-mitigated execution on a 156-qubit IBM Heron processor yields usable approximations to the Gibbs state of a 30-site transverse-field Ising model.
  • The framework directly supports thermal sampling tasks in quantum simulation and optimization.

Where Pith is reading between the lines

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

  • Off-loading free-energy evaluation to classical matrix product states can lower the quantum circuit depth needed for thermal-state preparation.
  • The same hybrid loop could be applied to other variational problems where classical pre-computation supplies a reliable cost function.
  • Success on present-day hardware suggests the method may serve as a testbed for studying how classical assistance affects convergence in variational quantum algorithms.

Load-bearing premise

The variational optimization of the hardware-efficient ansatz, guided by the classical MPS free-energy value, converges to a quantum state whose measured observables agree with the true Gibbs state within the reported error bars.

What would settle it

A measurement on the 30-site transverse-field Ising model that shows energy or susceptibility after mitigation deviating from the MPS-predicted value by more than the claimed 50 percent error reduction.

Figures

Figures reproduced from arXiv: 2510.23546 by Khadijeh Najafi, Rui-Hao Li, Semeon Valgushev.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the MPS-assisted variational Gibbs state preparation algorithm, showcasing two ansatzes benchmarked in [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Performance comparison of the TFDA and HEA on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Thermal energy estimates of the variationally prepared Gibbs states for 1D TFIM with (a) 20 and (b) 30 spins, and 2D [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Thermal estimates of magnetic susceptibility of the variationally prepared Gibbs states for 1D TFIM with (a) 20 and [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Thermal estimates of the two-point correlation func [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Thermal estimates of (a) energy and (b) magnetic [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

The preparation of quantum Gibbs states at finite temperatures is a cornerstone of quantum computation, enabling applications in quantum simulation of many-body systems, machine learning via quantum Boltzmann machines, and optimization through thermal sampling techniques. In this work, we introduce a variational framework that leverages matrix product states for the efficient classical evaluation of the Helmholtz free energy, combining scalable entanglement entropy computation with a hardware efficient ansatz to accurately approximate thermal states in one- and two-dimensional systems. We conduct extensive benchmarking on key observables, including energy density, susceptibility, specific heat, and two-point correlations, comparing against exact analytical results for 1D systems and quantum Monte Carlo simulations for 2D lattices across various temperatures and ansatz configurations. Our large-scale numerical simulations demonstrate the capability to prepare high-quality Gibbs states for 1D lattice models with up to 30 sites and 2D systems with up to 6x6 sites, using up to 44 qubits. Finally, we demonstrate the framework's practical viability on a 156-qubit IBM Heron processor by preparing the approximate Gibbs state of a 30-site transverse-field Ising model. Leveraging a combination of error mitigation techniques, we reduce the relative errors in energy and susceptibility measurements by over 50% compared to unmitigated results.

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 paper introduces a hybrid variational framework for preparing approximate quantum Gibbs states on digital quantum hardware. It employs classical matrix product states (MPS) to evaluate the Helmholtz free energy, guiding the optimization of parameters in a hardware-efficient ansatz. The approach is benchmarked via comparisons of energy density, susceptibility, specific heat, and correlation functions against exact 1D analytics and 2D quantum Monte Carlo results for lattices up to 30 sites (1D) and 6x6 (2D), using up to 44 qubits. A hardware demonstration prepares an approximate Gibbs state for a 30-site transverse-field Ising model on a 156-qubit IBM Heron processor, with error mitigation yielding over 50% reduction in relative errors for energy and susceptibility.

Significance. If the central claim holds, the work offers a practical route to thermal state preparation on near-term quantum devices by leveraging classical MPS for scalable free-energy guidance. This hybrid strategy could advance quantum simulation of finite-temperature many-body physics and related applications. Strengths include the scale of the numerical benchmarks (1D/2D systems with direct comparisons to exact/QMC data) and the real-hardware demonstration with mitigation; these provide concrete evidence of feasibility beyond small-system proofs of principle.

major comments (2)
  1. [Numerical Results / Benchmarking] The benchmarking sections (numerical simulations for 1D and 2D models) report close agreement between optimized observables and exact/QMC references, but provide limited analysis of ansatz expressivity or optimization convergence. Without diagnostics such as entanglement entropy matching the thermal value, fidelity estimates where feasible, or checks for multiple minima in the free-energy landscape, it remains possible that the hardware-efficient ansatz reaches states reproducing low-order observables within error bars while deviating from the true Gibbs state in higher moments or structure, particularly at lower temperatures. This directly affects the validity of the 'high-quality Gibbs states' claim.
  2. [Hardware Implementation] In the hardware demonstration section for the 30-site TFIM on the IBM Heron processor, the >50% error reduction via mitigation is presented as evidence of practical viability. However, the manuscript does not quantify how mitigation techniques (including any post-selection) alter the prepared state's proximity to the target thermal ensemble versus simply improving expectation values of the measured observables. This is load-bearing for interpreting the hardware results as successful Gibbs-state preparation rather than mitigated sampling of a variational state.
minor comments (2)
  1. [Abstract and Numerical Results] Clarify the precise qubit count and circuit depth for the 2D 6x6 simulations versus the 1D cases to avoid ambiguity in the 'up to 44 qubits' statement.
  2. [Method] The description of the MPS-assisted free-energy evaluation would benefit from an explicit equation or pseudocode showing how the entanglement entropy computation is combined with the variational parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments, which have helped us improve the clarity and rigor of the manuscript. We address each major comment in detail below and have made revisions where appropriate to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Numerical Results / Benchmarking] The benchmarking sections (numerical simulations for 1D and 2D models) report close agreement between optimized observables and exact/QMC references, but provide limited analysis of ansatz expressivity or optimization convergence. Without diagnostics such as entanglement entropy matching the thermal value, fidelity estimates where feasible, or checks for multiple minima in the free-energy landscape, it remains possible that the hardware-efficient ansatz reaches states reproducing low-order observables within error bars while deviating from the true Gibbs state in higher moments or structure, particularly at lower temperatures. This directly affects the validity of the 'high-quality Gibbs states' claim.

    Authors: We agree that additional diagnostics would strengthen the evidence for the quality of the approximated Gibbs states. In the revised manuscript, we have added a new subsection discussing the entanglement entropy of the optimized variational states for the 1D models, demonstrating close agreement with the exact thermal entanglement entropy obtained from the transfer-matrix method. We also include convergence plots of the free-energy optimization for representative temperatures and system sizes, showing consistent convergence across multiple random initial parameter sets with no evidence of trapping in significantly suboptimal minima. While direct fidelity estimation remains intractable for the largest systems considered, the combination of agreement on energy, susceptibility, specific heat, and correlation functions with the entropy diagnostic supports that the states capture the essential structure of the thermal ensemble beyond low-order observables. These additions are now reflected in the updated numerical results and discussion sections. revision: yes

  2. Referee: [Hardware Implementation] In the hardware demonstration section for the 30-site TFIM on the IBM Heron processor, the >50% error reduction via mitigation is presented as evidence of practical viability. However, the manuscript does not quantify how mitigation techniques (including any post-selection) alter the prepared state's proximity to the target thermal ensemble versus simply improving expectation values of the measured observables. This is load-bearing for interpreting the hardware results as successful Gibbs-state preparation rather than mitigated sampling of a variational state.

    Authors: We acknowledge the importance of this distinction. Because full quantum state tomography or direct fidelity computation to the target Gibbs state is infeasible on current hardware for a 30-site system, we cannot provide a quantitative metric of the prepared state's distance to the ideal thermal ensemble. The variational optimization itself is performed classically via MPS evaluation of the free energy, and the hardware step serves to prepare the resulting ansatz and extract observables. In the revised manuscript, we have clarified the role of error mitigation: it is applied post-preparation to reduce bias and variance in the measured expectation values, thereby enabling a more reliable comparison to the classically optimized targets. We have added an explicit discussion of this limitation, noting that hardware noise affects the fidelity of state preparation but that the mitigation improves the accuracy of the extracted observables without altering the underlying variational guarantee in the ideal case. This revised framing better contextualizes the hardware results as a demonstration of practical measurement rather than a direct claim of enhanced state fidelity. revision: partial

Circularity Check

0 steps flagged

No significant circularity; claims rest on external benchmarks and hardware execution

full rationale

The paper optimizes a hardware-efficient ansatz via classical MPS evaluation of the Helmholtz free energy, then validates the resulting quantum state by direct comparison of observables (energy density, susceptibility, specific heat, correlations) to exact 1D solutions and independent QMC simulations for 2D lattices up to 6x6. The 30-site TFIM demonstration on the 156-qubit IBM Heron processor further uses real hardware execution with error mitigation. These external references and the separation between classical MPS free-energy computation and quantum variational preparation mean the central claims do not reduce to self-definition, fitted-input renaming, or self-citation chains. The derivation chain is self-contained against independent benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the ability of MPS to efficiently compute free energy and entropy for the target systems and on the hardware-efficient ansatz being sufficiently expressive for the temperatures studied.

free parameters (2)
  • ansatz variational parameters
    Parameters of the hardware-efficient circuit are optimized to minimize the free energy.
  • MPS bond dimension
    Truncation parameter controlling classical representation accuracy.
axioms (1)
  • domain assumption Matrix product states provide an efficient and accurate classical representation for computing the Helmholtz free energy and entanglement entropy of the thermal states considered.
    Invoked to justify the classical precomputation step for 1D and 2D lattices.

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