Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2610.40361

Rapid mixing for Gibbs samplers for quantum spin systems with all-to-all interactions

Nicola Bertoni, Giacomo De Palma

quant-phSubmitted Oct 29, 2026• Announced Oct 30, 2026• 0 citations
arXivHTMLPDF

Abstract

Understanding when quantum many-body systems rapidly converge to thermal equilibrium is a central problem in quantum statistical mechanics. While spatial locality provides powerful tools for systems with an underlying geometry, fully connected models require estimates that accommodate interactions between every pair of spins. We study Gibbs-state preparation for two-local quantum Hamiltonians on arbitrary interaction graphs, including fully connected systems, via Lindblad dynamics that satisfies Kubo--Martin--Schwinger (KMS) detailed balance. Our approach is based on a quantum analogue of the classical Dobrushin condition, which controls the influence of local perturbations on the rest of the system. Using the quantum Wasserstein distance of order $1$, we quantify how these perturbations propagate under the dynamics and derive contraction estimates in a high-temperature regime controlled by the maximum total interaction strength per site. These estimates require no assumptions on spatial geometry or graph growth and yield mixing times logarithmic in the system size and the inverse error tolerance, uniformly over all initial states.

AI Summary

The paper asks when noncommuting quantum spin systems on arbitrary interaction graphs, including complete graphs, admit logarithmic-time Gibbs sampling without geometric locality or graph-growth assumptions. It studies an existing KMS-detailed-balanced Lindbladian with imaginary-time-dressed single-site Pauli jumps and a coherent correction, targeting $\sigma=e^{-\beta H}/\operatorname{tr}(e^{-\beta H})$ for two-body qubit Hamiltonians $H=\sum_{i<j}H_{i,j}$. The controlling parameter is the maximum incident interaction strength, $\Lambda=\max_i\sum_{j\ne i}\|H_{i,j}\|$, for the prescribed decomposition. If $\beta\Lambda<1/100$, then every initial state satisfies $\|e^{t\mathcal L}(\rho)-\sigma\|_{\mathrm{tr}}\le\varepsilon$ whenever $t\ge\frac13\log(2n/\varepsilon)$. Time uses the extensive generator $\mathcal L=\sum_i\mathcal L_i$, not a generator normalized by $n$. For complete-graph interactions scaled by $1/\sqrt{n-1}$ with unscaled pair norms at most $J$, the condition becomes $\beta<1/(100J\sqrt{n-1})$; mean-field scaling by $1/(n-1)$ permits $\beta<1/(100J)$ independently of size. The proof verifies a quantum Dobrushin contraction criterion in quantum Wasserstein distance of order one. Weighted connected-cluster expansions and marked-site sums control imaginary-time dressing using $\Lambda$. An exactly reversible lift of real-time evolution to transport plans controls the Fourier-represented coherent correction while retaining zeroth-order cancellations. These estimates preserve a uniform contraction rate from the infinite-temperature depolarizing dynamics. The guarantees concern this specified sampler, two-body qubit interactions, and sufficiently high temperature. They are not uniform in arbitrary on-site fields. Circuit complexity additionally requires bounds on implementation cost; logarithmic continuous-time mixing alone does not provide them.

Little sign of AI-generated textAI use disclosed
An AI-assisted language model was used during manuscript preparation for literature discovery, editorial restructuring, and consistency checks.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

Related papers

Rapid mixing of Gibbs samplers via quantum Dobrushin--Shlosman conditions

Cambyse Rouzé, Daniel Stilck França

Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error $\varepsilon$ using $N\operatorname{polylog}(N/\varepsilon)$ gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.

quant-pharXiv:2610.03578•Oct 5, 2026•0 citations

Spectral Analysis Beyond the Spectral Gap: Rapid Mixing of Quantum Gibbs Samplers in One Dimension

Hongrui Chen, Zhiyan Ding

We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of $n$ sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error $ε$ in time $O(\log^4(n)+\log(1/ε))$. This improves the previous $O(n+\log(1/ε))$ bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate $Ω(\log^{-4}(n))$.

quant-pharXiv:2610.05848•Oct 6, 2026•0 citations

Overcoming the Lamb Shift in System-Bath Interaction Models via KMS Detailed Balance: High-Accuracy Thermalization with Time-Bounded Interactions

Hongrui Chen, Zhiyan Ding, Ruizhe Zhang

We investigate quantum thermal state preparation algorithms based on system-bath interactions and uncover a surprising phenomenon in the weak-coupling regime. We rigorously prove that, if the system-bath interaction is engineered so that the transition part of the approximate Lindbladian generator satisfies the Kubo--Martin--Schwinger (KMS) detailed balance condition, then the unique fixed point of the dynamics can be made arbitrarily close to the Gibbs state in the weak-coupling limit, regardless of the structure of the Lamb shift term. Importantly, this remains true even when the approximate Lindbladian differs substantially from the ideal Davies generator and the Lamb shift term does not commute with the thermal state. Our result shows that the role of the KMS detailed balance condition extends well beyond standard Lindbladian dynamics, serving as a general principle for a broader class of dissipative systems. Furthermore, by combining this with a general perturbation framework, we bound the mixing time of the dynamics and establish an end-to-end complexity of $\mathcal{O}(\varepsilon^{-1})$ for Gibbs state preparation. These guarantees apply to any Hamiltonian whose associated KMS-detailed-balance Lindbladian is known to be fast mixing.

quant-pharXiv:2604.15616•Apr 20, 2026•0 citations

Fast convergence of propagation algorithms for open quantum systems

Giorgio Facelli, Hamza Fawzi, Omar Fawzi +3 more

We establish general conditions for the rapid convergence of propagation algorithms for computing expectation values of local observables in quantum many-body systems, covering both spin and fermionic systems. Our results apply to evolutions that are sufficiently contractive, providing a general mechanism by which contractivity controls the growth generated by local interactions and enables efficient classical simulation. As a first application, we consider noisy time evolution generated by local Hamiltonians on arbitrary interaction graphs. We show that the dynamics can be efficiently simulated when the noise strength $λ$ is sufficiently large compared to the degree of the interaction graph. More generally, for systems with an interaction strength $u$, our bounds determine a time horizon $t_{\text{max}}$ as a function of the interaction strength $u$, the noise strength $λ$, and the interaction structure below which the algorithm is efficient. In the noiseless limit, $λ=0$, our analysis extends the efficiently simulable time scale from $\log(1/u)$ proved in Facelli, Fawzi, and Fawzi (2026) to $t_{\text{max}} \sim 1/u$ matching the recent improvement by Zhao, Marvian and Tong (2026). As a second application, we apply the same framework to the computation of local observables in Gibbs states of local Hamiltonians. We consider a pseudo-Lindbladian whose steady state is the Gibbs state and show that, at sufficiently high temperature, its propagation algorithm converges rapidly. In particular, for weakly-interacting systems with interaction strength $u$, our result applies up to inverse temperatures $β\sim\log(1/u)$.

quant-pharXiv:2610.06769•Oct 6, 2026•0 citations

Efficient Quantum Gibbs Samplers with Kubo–Martin–Schwinger Detailed Balance Condition

Zhiyan Ding, Bowen Li, Lin Lin

Abstract Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.

arXiv:2404.05998•Communications in Mathematical Physics•Feb 27, 2025•35 citations

Rapid mixing of quantum spin chains at any finite temperature

Leeseok Kim

We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in $\mathcal{O}\left(\log (n/\varepsilon)\right)$ time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order $1$. We further provide $\mathcal{O}(\text{polylog}(n/\varepsilon))$ depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.

quant-pharXiv:2610.01190•Oct 2, 2026•0 citations
Understanding when quantum many-body systems rapidly converge to thermal equilibrium is a central problem in quantum statistical mechanics. While spatial locality provides powerful tools for systems with an underlying geometry, fully connected models require estimates that accommodate interactions between every pair of spins. We study Gibbs-state preparation for two-local quantum Hamiltonians on arbitrary interaction graphs, including fully connected systems, via Lindblad dynamics that satisfies Kubo--Martin--Schwinger (KMS) detailed balance. Our approach is based on a quantum analogue of the classical Dobrushin condition, which controls the influence of local perturbations on the rest of the system. Using the quantum Wasserstein distance of order 111, we quantify how these perturbations propagate under the dynamics and derive contraction estimates in a high-temperature regime controlled by the maximum total interaction strength per site. These estimates require no assumptions on spatial geometry or graph growth and yield mixing times logarithmic in the system size and the inverse error tolerance, uniformly over all initial states.
The paper asks when noncommuting quantum spin systems on arbitrary interaction graphs, including complete graphs, admit logarithmic-time Gibbs sampling without geometric locality or graph-growth assumptions. It studies an existing KMS-detailed-balanced Lindbladian with imaginary-time-dressed single-site Pauli jumps and a coherent correction, targeting σ=e−βH/tr⁡(e−βH)\sigma=e^{-\beta H}/\operatorname{tr}(e^{-\beta H})σ=e−βH/tr(e−βH) for two-body qubit Hamiltonians H=∑i<jHi,jH=\sum_{i<j}H_{i,j}H=∑i<j​Hi,j​.
The controlling parameter is the maximum incident interaction strength, Λ=max⁡i∑j≠i∥Hi,j∥\Lambda=\max_i\sum_{j\ne i}\|H_{i,j}\|Λ=maxi​∑j=i​∥Hi,j​∥, for the prescribed decomposition. If βΛ<1/100\beta\Lambda<1/100βΛ<1/100, then every initial state satisfies ∥etL(ρ)−σ∥tr≤ε\|e^{t\mathcal L}(\rho)-\sigma\|_{\mathrm{tr}}\le\varepsilon∥etL(ρ)−σ∥tr​≤ε whenever t≥13log⁡(2n/ε)t\ge\frac13\log(2n/\varepsilon)t≥31​log(2n/ε). Time uses the extensive generator L=∑iLi\mathcal L=\sum_i\mathcal L_iL=∑i​Li​, not a generator normalized by nnn. For complete-graph interactions scaled by 1/n−11/\sqrt{n-1}1/n−1​ with unscaled pair norms at most JJJ, the condition becomes β<1/(100Jn−1)\beta<1/(100J\sqrt{n-1})β<1/(100Jn−1​); mean-field scaling by 1/(n−1)1/(n-1)1/(n−1) permits β<1/(100J)\beta<1/(100J)β<1/(100J) independently of size.
The proof verifies a quantum Dobrushin contraction criterion in quantum Wasserstein distance of order one. Weighted connected-cluster expansions and marked-site sums control imaginary-time dressing using Λ\LambdaΛ. An exactly reversible lift of real-time evolution to transport plans controls the Fourier-represented coherent correction while retaining zeroth-order cancellations. These estimates preserve a uniform contraction rate from the infinite-temperature depolarizing dynamics.
The guarantees concern this specified sampler, two-body qubit interactions, and sufficiently high temperature. They are not uniform in arbitrary on-site fields. Circuit complexity additionally requires bounds on implementation cost; logarithmic continuous-time mixing alone does not provide them.
Classical Dobrushin--Shlosman theory uses heat-bath block updates to identify sharp temperature thresholds for establishing the rapid mixing of Gibbs samplers. In particular, these mixing guarantees often hold even when stricter single-site conditions fail. We extend this approach to noncommuting quantum lattice systems through a quantum Dobrushin--Shlosman condition for finite-block dynamics, which we call smoothed heat-bath dynamics. We establish this condition in two regimes. For finite-range quantum spin chains at every fixed finite temperature, we prove logarithmic trace-norm mixing in system size, with normalized bounds uniform in on-site fields and single exponential in inverse temperature. We also prove stability under small local perturbations of interacting, possibly noncommuting reference Hamiltonians on graphs of polynomial volume growth. For classical references with bounded local interactions, uniform strong spatial mixing implies our quantum condition. With bounded local terms and fixed physical, geometric and block parameters, the resulting samplers prepare Gibbs states to trace-norm error ε\varepsilonε using Npolylog⁡(N/ε)N\operatorname{polylog}(N/\varepsilon)Npolylog(N/ε) gates and classical operations. Our block updates combine a local Gibbs reset with quantum belief propagation to incorporate interactions across the block boundary, yielding exactly Gibbs-preserving, KMS-reversible channels. Two independently tunable parameters control contraction: an internal evolution time suppresses the influence of sites inside a block, while the block size allows interior contraction to dominate boundary influence. This finite-time relaxation has no counterpart in a classical heat-bath block update.
We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of nnn sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error εεε in time O(log⁡4(n)+log⁡(1/ε))O(\log^4(n)+\log(1/ε))O(log4(n)+log(1/ε)). This improves the previous O(n+log⁡(1/ε))O(n+\log(1/ε))O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log⁡−4(n))Ω(\log^{-4}(n))Ω(log−4(n)).
We investigate quantum thermal state preparation algorithms based on system-bath interactions and uncover a surprising phenomenon in the weak-coupling regime. We rigorously prove that, if the system-bath interaction is engineered so that the transition part of the approximate Lindbladian generator satisfies the Kubo--Martin--Schwinger (KMS) detailed balance condition, then the unique fixed point of the dynamics can be made arbitrarily close to the Gibbs state in the weak-coupling limit, regardless of the structure of the Lamb shift term. Importantly, this remains true even when the approximate Lindbladian differs substantially from the ideal Davies generator and the Lamb shift term does not commute with the thermal state. Our result shows that the role of the KMS detailed balance condition extends well beyond standard Lindbladian dynamics, serving as a general principle for a broader class of dissipative systems. Furthermore, by combining this with a general perturbation framework, we bound the mixing time of the dynamics and establish an end-to-end complexity of O(ε−1)\mathcal{O}(\varepsilon^{-1})O(ε−1) for Gibbs state preparation. These guarantees apply to any Hamiltonian whose associated KMS-detailed-balance Lindbladian is known to be fast mixing.
We establish general conditions for the rapid convergence of propagation algorithms for computing expectation values of local observables in quantum many-body systems, covering both spin and fermionic systems. Our results apply to evolutions that are sufficiently contractive, providing a general mechanism by which contractivity controls the growth generated by local interactions and enables efficient classical simulation. As a first application, we consider noisy time evolution generated by local Hamiltonians on arbitrary interaction graphs. We show that the dynamics can be efficiently simulated when the noise strength λλλ is sufficiently large compared to the degree of the interaction graph. More generally, for systems with an interaction strength uuu, our bounds determine a time horizon tmaxt_{\text{max}}tmax​ as a function of the interaction strength uuu, the noise strength λλλ, and the interaction structure below which the algorithm is efficient. In the noiseless limit, λ=0λ=0λ=0, our analysis extends the efficiently simulable time scale from log⁡(1/u)\log(1/u)log(1/u) proved in Facelli, Fawzi, and Fawzi (2026) to tmax∼1/ut_{\text{max}} \sim 1/utmax​∼1/u matching the recent improvement by Zhao, Marvian and Tong (2026). As a second application, we apply the same framework to the computation of local observables in Gibbs states of local Hamiltonians. We consider a pseudo-Lindbladian whose steady state is the Gibbs state and show that, at sufficiently high temperature, its propagation algorithm converges rapidly. In particular, for weakly-interacting systems with interaction strength uuu, our result applies up to inverse temperatures β∼log⁡(1/u)β\sim\log(1/u)β∼log(1/u).
Abstract Lindblad dynamics and other open-system dynamics provide a promising path towards efficient Gibbs sampling on quantum computers. In these proposals, the Lindbladian is obtained via an algorithmic construction akin to designing an artificial thermostat in classical Monte Carlo or molecular dynamics methods, rather than being treated as an approximation to weakly coupled system-bath unitary dynamics. Recently, Chen, Kastoryano, and Gilyén (arXiv:2311.09207) introduced the first efficiently implementable Lindbladian satisfying the Kubo–Martin–Schwinger (KMS) detailed balance condition, which ensures that the Gibbs state is a fixed point of the dynamics and is applicable to non-commuting Hamiltonians. This Gibbs sampler uses a continuously parameterized set of jump operators, and the energy resolution required for implementing each jump operator depends only logarithmically on the precision and the mixing time. In this work, we build upon the structural characterization of KMS detailed balanced Lindbladians by Fagnola and Umanità, and develop a family of efficient quantum Gibbs samplers using a finite set of jump operators (the number can be as few as one), akin to the classical Markov chain-based sampling algorithm. Compared to the existing works, our quantum Gibbs samplers have a comparable quantum simulation cost but with greater design flexibility and a much simpler implementation and error analysis. Moreover, it encompasses the construction of Chen, Kastoryano, and Gilyén as a special instance.
We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in O(log⁡(n/ε))\mathcal{O}\left(\log (n/\varepsilon)\right)O(log(n/ε)) time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order 111. We further provide O(polylog(n/ε))\mathcal{O}(\text{polylog}(n/\varepsilon))O(polylog(n/ε)) depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.