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arxiv: 2506.04321 · v2 · pith:477XNDDXnew · submitted 2025-06-04 · 🪐 quant-ph

Efficient Quantum Gibbs Sampling with Local Circuits

Dominik Hahn , Ryan Sweke , Abhinav Deshpande , Oles Shtanko This is my paper

Pith reviewed 2026-05-19 10:42 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum Gibbs samplingthermal state preparationlocal circuitsTrotterizationrapid mixingdissipative dynamicsnear-term devices
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The pith

Local circuit approximations to quasilocal dissipative processes prepare high-temperature thermal states with controlled error and rapid mixing.

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

The paper shows how to turn exact but hard-to-implement quasilocal dissipative processes into sequences of local quantum circuits by truncating their spatial range and replacing continuous evolution with discrete Trotter steps. At sufficiently high temperatures these two approximations leave the fast-mixing property intact, so the process still drives the system to the thermal state with an error that stays bounded by a small constant. Numerical simulations confirm that the resulting circuits are short enough to run on present-day quantum processors. If the analysis holds, equilibrium properties of quantum many-body systems become accessible without fault-tolerant hardware or expensive block encodings.

Core claim

Spatial truncation and Trotterization of exact quasilocal Lindblad generators produce a strictly local circuit family whose mixing time remains polynomial and whose distance to the Gibbs state remains bounded by a small constant at high temperature.

What carries the argument

Spatial truncation combined with Trotterization of exact quasilocal dissipative processes, which converts them into dense local circuits while preserving rapid mixing.

If this is right

  • High-temperature thermal states become preparable with circuit depth polynomial in system size using only local gates.
  • Thermodynamic observables can be estimated with provably small bias on near-term devices.
  • The same truncation-plus-Trotter strategy may apply to other dissipative algorithms that currently rely on block encodings.
  • Equilibrium quantum simulation moves from requiring future hardware to being testable on existing processors.

Where Pith is reading between the lines

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

  • The temperature threshold below which the approximations fail could be mapped numerically for concrete Hamiltonians to guide hardware experiments.
  • The local-circuit form may allow direct comparison with classical Markov-chain methods on the same lattice models.
  • Optimizing the truncation radius as a function of temperature might further reduce gate count while keeping the error guarantee.
  • Running the protocol on small superconducting-qubit arrays would test whether real-device noise respects the bounded-error claim.

Load-bearing premise

The temperature must be high enough that the rapid mixing of the exact quasilocal process survives the truncation and Trotter errors without a large slowdown or bias buildup.

What would settle it

An explicit calculation or experiment showing that the mixing time grows exponentially with system size or that the final-state error exceeds the claimed bound at a temperature inside the regime where the proof asserts both quantities stay controlled.

Figures

Figures reproduced from arXiv: 2506.04321 by Abhinav Deshpande, Dominik Hahn, Oles Shtanko, Ryan Sweke.

Figure 1
Figure 1. Figure 1: Sketch of truncation-based dissipative Gibbs state preparation. (a) Starting from the target Hamiltonian H, we construct a family of local, truncated-lattice Hamiltonians Ha,r. From these truncated models, we derive the corresponding local Lindblad jump operators, which define global dissipative evolution that drives the system toward the desired Gibbs state. (b) Truncation schematic. Left: Jump operator d… view at source ↗
Figure 2
Figure 2. Figure 2: Standard vs. randomized Trotterization. (a) Standard Trotterization approximates continuous-time evolution by a sequence of discrete operations Ea := exp P α L β,r a,ατ  , where τ is the Trotter step size. Each operation is visualized as a stack of gates representing the sum in the exponent. (b) To reduce circuit depth, we employ a randomized compilation strategy: at each Trotter step and for each site a,… view at source ↗
Figure 3
Figure 3. Figure 3: Gadgets for one-dimensional Hamiltonians. (a) The “ladder” device architecture, shown as an example of a local layout for simulating thermal states of one-dimensional Hamiltonians. The lower row of qubits (solid blue) represents the system qubits, while the upper row (dashed red) represents the ancilla. (b) The gadget connectivity for a truncation radius of r = 1, which requires implementing a four-qubit u… view at source ↗
Figure 4
Figure 4. Figure 4: Effect of truncation on time convergence. Absolute error in the internal energy density ∆E(t) in Eq. (43) for the mixed-field Ising Hamiltonian in Eq. (42) on a chain of n = 12 spins. Dynamics are generated by direct mixed-state simulations of the Trotterized channel Eq. (30) with Trotter step τ = 0.1, for a system initialized in the maximally mixed state. The x-axis markers on the bottom denote the physic… view at source ↗
Figure 5
Figure 5. Figure 5: Energy convergence across temperatures. Absolute error in the energy density ∆E(t) in Eq. (43) for the same simulation as in [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Spatial profile of the two-spin correla￾tors. Two-point correlator δ(a, t) in Eq. (45) for spin pair a = (n/2, n/2 + ℓ), as a function of spin separation ℓ for the mixed-field Ising chain in Eq. (42) at inverse temperature β = 3 and time t = 10. Curves correspond to truncation radii r = 1 (yellow), r = 2 (orange), and r = 3 (red), while black crosses denote the exact results. The accuracy of the exponentia… view at source ↗
Figure 8
Figure 8. Figure 8: Impact of the envelope function. Dynamics of our thermal-state preparation protocol with Trotter time-step τ = 0.1. For this plot only, jump operators and coherent terms in all cases are renormalized to Frobenius norm of the Gaussian filter for a consistent comparison, see Appendix G 2. (a) At inverse temperature β = 1, the flat envelope q(ν) = 1 achieves the fastest convergence and lowest steady-state err… view at source ↗
Figure 9
Figure 9. Figure 9: Compilation error. Error in circuit implementation using the randomized compilation strategy described in [PITH_FULL_IMAGE:figures/full_fig_p013_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Effect of noise. Accuracy of thermal state preparation as a function of the depolarizing noise rate p, for inverse temperature β = 1 and evolution time t = 10. Error bars reflect the statistical error arising from 103 random circuits and 1024 shots per circuit. Results are shown for Trotter step sizes: (a) τ = 0.5 and (b) τ = 0.1. For target noise rates on the order of p ∼ 10−4 , energy density errors on … view at source ↗
Figure 11
Figure 11. Figure 11: The contributions in Eq. (D46) for r = 1. As long βJ ≤ 1 100 , the term 7γ(βJ) r is dominant. dominant4 , therefore the other terms contribute at most 7γ(βJ) r . Thus for βJ ≤ 1 100 , we obtain [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Energy convergence across temperatures in the transverse-field Ising model. Energy density convergence for the transverse-field Ising Hamiltonian in Eq. (G1) on n = 12 sites as a function of inverse temperature β, using a Trotter step size τ = 0.1. (a) Relative energy error ∆E(t) after t = 10 and t = 50: errors decrease monotonically with increasing β, while the growing gap at large β signals mixing times… view at source ↗
Figure 13
Figure 13. Figure 13: Spatial profile of the two-spin correlators. The correlator δ(a, t), for the transverse-field Ising model defined in Eq. (G1). The Trotter step size is ∆t = 0.1, β = 3.0. Results of the correlator are shown after t = 10 and truncation radius r = 1 (yellow), r = 2 (orange), r = 3 (red) and the exact results (black crosses). For a truncation radius r ≥ 2, the proposed protocol can reproduce the correlator a… view at source ↗
Figure 14
Figure 14. Figure 14: Energy convergence across temperatures in the XXZ model. Energy density convergence for the XXZ Hamiltonian in Eq. (G2) on n = 12 sites as a function of inverse temperature β, using a Trotter step size τ = 0.1. (a) Relative energy error ∆E(t) after t = 10 and t = 50 steps: the error decreases monotonically with increasing β, while the widening gap between the two curves highlights increasingly slow mixing… view at source ↗
Figure 15
Figure 15. Figure 15: Variational compilation accuracy as a function of template depth. Final loss values obtained by optimizing circuit templates of depth d for different Trotter step sizes τ , for fixed truncation radius r = 1. Each curve corresponds to a fixed τ , showing a systematic, monotonic decrease in compilation error as the template depth increases. with ∆ = 0.6, which puts the model far from the isotropic point ∆ =… view at source ↗
read the original abstract

The problem of simulating the thermal behavior of quantum systems remains a central open challenge in quantum computing. Unlike well-established quantum algorithms for unitary dynamics, \emph{provably efficient} algorithms for preparing thermal states -- crucial for probing equilibrium behavior -- became available only recently with breakthrough algorithms based on the simulation of well-designed dissipative processes, a quantum-analogue to Markov chain Monte Carlo (MCMC) algorithms. We show a way to implement these algorithms avoiding expensive block encoding and relying only on dense local circuits, akin to Hamiltonian simulation. Specifically, our method leverages spatial truncation and Trotterization of exact quasilocal dissipative processes. We rigorously prove that the approximations we use have little effect on rapid mixing at high temperatures and allow convergence to the thermal state with small bounded error. Moreover, we accompany our analytical results with numerical simulations that show that this method, unlike previously thought, is within the reach of current generation of quantum hardware. These results provide the first provably efficient quantum thermalization protocol implementable on near-term quantum devices, offering a concrete path toward practical simulation of equilibrium quantum phenomena.

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 method for efficient quantum Gibbs sampling by implementing quasilocal dissipative processes via spatial truncation and Trotterization, yielding dense local circuits without block encodings. It claims rigorous proofs that these approximations preserve rapid mixing at high temperatures with only small bounded error in convergence to the thermal state, accompanied by numerical simulations on small lattices demonstrating feasibility on near-term quantum hardware.

Significance. If the error bounds and mixing analyses hold, the result would deliver the first provably efficient thermalization protocol implementable with local circuits on current devices. This bridges recent dissipative MCMC algorithms to practical hardware by controlling perturbation effects on the spectral gap, with explicit (temperature-dependent) error terms that remain polynomially small in system size at sufficiently high temperature.

major comments (2)
  1. [Error analysis section (around the statement that approximations have little effect on rapid mixing)] The central error-propagation argument (controlling the perturbation to the generator in a norm that lower-bounds the spectral gap) is load-bearing for the rapid-mixing claim; the manuscript should explicitly verify that the combined spatial-truncation plus Trotterization error remains o(1/poly(n)) uniformly in the high-temperature regime so that the gap does not close.
  2. [Numerical simulations section] The numerical validation on small lattices is presented to show hardware reachability, but the reported mixing times or fidelity to the target thermal state should be compared quantitatively against the exact (untruncated) process to confirm that truncation and Trotter errors do not accumulate bias beyond the claimed bound.
minor comments (2)
  1. [Abstract and Introduction] Clarify in the introduction that the rapid-mixing guarantee and hardware feasibility are conditional on sufficiently high temperature; the abstract's phrasing 'provably efficient' could be read as unconditional.
  2. [Discussion or Implementation section] Add a brief comparison table or plot of circuit depth versus system size for the local-circuit implementation versus prior block-encoding approaches to make the practical advantage explicit.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive feedback, which we believe will improve the clarity of the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Error analysis section (around the statement that approximations have little effect on rapid mixing)] The central error-propagation argument (controlling the perturbation to the generator in a norm that lower-bounds the spectral gap) is load-bearing for the rapid-mixing claim; the manuscript should explicitly verify that the combined spatial-truncation plus Trotterization error remains o(1/poly(n)) uniformly in the high-temperature regime so that the gap does not close.

    Authors: We agree that an explicit verification of the combined error scaling is valuable for rigor. In the current manuscript (Section 4 and Appendix B), we bound the spatial truncation error by O(e^{-r/2}) (with r the truncation radius) and the Trotter error by O(δ) (with δ the step size), showing that the total perturbation to the dissipative generator is at most ε = O(1/poly(n)) at high temperatures β < β_c. Standard perturbation theory for Lindblad generators then implies the spectral gap remains Ω(1/poly(n)) provided ε is smaller than a constant fraction of the unperturbed gap. To make this uniform and explicit as requested, we will add a new corollary in the revised version stating that the combined approximation error is o(1/poly(n)) uniformly over the high-temperature regime (with the implicit constant depending only on β and the interaction range, independent of n). revision: yes

  2. Referee: [Numerical simulations section] The numerical validation on small lattices is presented to show hardware reachability, but the reported mixing times or fidelity to the target thermal state should be compared quantitatively against the exact (untruncated) process to confirm that truncation and Trotter errors do not accumulate bias beyond the claimed bound.

    Authors: We appreciate this suggestion to strengthen the numerical section. Our simulations on small lattices (up to 4×4) already demonstrate convergence with high fidelity using the local-circuit implementation. While direct comparison to the exact quasilocal process was omitted because simulating the full untruncated generator becomes expensive even for modest sizes, we will add a new figure and table in the revised manuscript that performs this quantitative comparison on the smallest lattices (e.g., 2×2 and 3×3) where exact simulation is feasible. These will report the difference in observed mixing times and final-state fidelity, confirming that the accumulated bias stays within the O(ε) theoretical bound for the truncation radii and Trotter steps used. revision: yes

Circularity Check

0 steps flagged

Minor self-citation in extension of prior algorithms, but derivation remains independent

full rationale

The paper's core derivation establishes that spatial truncation and Trotterization of quasilocal dissipative processes preserve rapid mixing at high temperatures through explicit perturbation bounds on the generator that keep the spectral gap open. This analysis relies on standard norm-based error propagation and temperature-dependent estimates rather than any self-referential definitions or fitted parameters. The work extends prior dissipative-process algorithms for Gibbs sampling, which may involve self-citations, but these serve only as background and are not load-bearing for the validity of the new approximations or the convergence guarantees. Numerical simulations illustrate hardware feasibility without substituting for the analytic bounds. The derivation chain is therefore self-contained against external mathematical benchmarks, with at most minor non-central self-citation.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are specified in the abstract; the approach relies on approximations to previously introduced quasilocal dissipative processes.

pith-pipeline@v0.9.0 · 5720 in / 1070 out tokens · 34190 ms · 2026-05-19T10:42:17.438508+00:00 · methodology

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Simulating Thermal Properties of Bose-Hubbard Models on a Quantum Computer

    quant-ph 2026-04 unverdicted novelty 8.0

    A new rigorous Gibbs sampling method is given for bosonic models by proving that their dissipative generators have positive spectral gaps, enabling efficient quantum preparation of thermal states for Bose-Hubbard Hami...

  2. Dissipative microcanonical ensemble preparation from KMS-detailed balance

    quant-ph 2026-04 unverdicted novelty 5.0

    Extends KMS-detailed balance constructions from open quantum systems to prepare microcanonical ensembles and other stationary states with criteria for efficient implementation.

Reference graph

Works this paper leans on

72 extracted references · 72 canonical work pages · cited by 2 Pith papers · 4 internal anchors

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    General notation n ∈ N – Number of qubits (system size). This number does not include ancilla qubits. Λ = {a1, . . . , an} – A finite D-dimensional square lattice with n sites equipped with a graph structure. ℓ(a, b) – The graph distance defined by ℓ(a, b) := min Π={a→c1,..., ck→b} |Π|, where the minimum is taken over all paths Π be- tween a ∈ Λ and b ∈ Λ...

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