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Quantum generalizations of Glauber and Metropolis dynamics

András Gilyén, Chi-Fang Chen, Joao F. Doriguello, Michael J. Kastoryano

TL;DR

This work develops discrete-time quantum Gibbs samplers that achieve exact $m{ ho}$-detailed balance, preserve locality, and are efficiently implementable on quantum hardware. By unifying classical Metropolis/Glauber ideas with quantum detailed balance through linear superoperators, the authors introduce coherent reweighting and coherent-rejection schemes that yield exactly detailed-balanced CP maps and channels, alongside alternative constructions that interpolate toward Davies-type dynamics. They prove that the discrete-time constructions maintain or exceed the spectral gaps of their continuous-time counterparts in the high-temperature regime and establish ergodicity-preserving properties for the resulting dynamics. The paper also details practical implementation routes via block-encoding and quantum singular value transformation, analyzes mixing times, and discusses the physical grounding of these quantum MCMC analogues within open quantum system frameworks, while outlining open questions about non-self-adjoint transitions and potential quadratic-speedup avenues.

Abstract

Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with many degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gilyén [CKG23] proposed a continuous-time quantum thermodynamic analogue to Glauber dynamics that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction resembles the dissipative dynamics arising from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time counterpart to any continuous-time quantum Gibbs sampler. Our construction preserves the desirable features (i)-(iii) while does not decrease the spectral gap. Also, we give an alternative highly coherent quantum generalization of detailed balanced dynamics that resembles another physically derived master equation, and propose a smooth interpolation between this and earlier constructions. Moreover, we show how to make earlier Metropolis-style Gibbs samplers (which estimate energies both before and after jumps) exactly detailed balanced. We study generic properties of all constructions, including the uniqueness of the fixed point, the (quasi-)locality of the resulting operators. Finally, we prove that the spectral gap of our new highly coherent Gibbs sampler is constant at high temperatures, thereby it mixes fast. We hope that our systematic approach to quantum Glauber and Metropolis dynamics will lead to widespread applications in various domains.

Quantum generalizations of Glauber and Metropolis dynamics

TL;DR

This work develops discrete-time quantum Gibbs samplers that achieve exact -detailed balance, preserve locality, and are efficiently implementable on quantum hardware. By unifying classical Metropolis/Glauber ideas with quantum detailed balance through linear superoperators, the authors introduce coherent reweighting and coherent-rejection schemes that yield exactly detailed-balanced CP maps and channels, alongside alternative constructions that interpolate toward Davies-type dynamics. They prove that the discrete-time constructions maintain or exceed the spectral gaps of their continuous-time counterparts in the high-temperature regime and establish ergodicity-preserving properties for the resulting dynamics. The paper also details practical implementation routes via block-encoding and quantum singular value transformation, analyzes mixing times, and discusses the physical grounding of these quantum MCMC analogues within open quantum system frameworks, while outlining open questions about non-self-adjoint transitions and potential quadratic-speedup avenues.

Abstract

Classical Markov Chain Monte Carlo methods have been essential for simulating statistical physical systems and have proven well applicable to other systems with many degrees of freedom. Motivated by the statistical physics origins, Chen, Kastoryano, and Gilyén [CKG23] proposed a continuous-time quantum thermodynamic analogue to Glauber dynamics that is (i) exactly detailed balanced, (ii) efficiently implementable, and (iii) quasi-local for geometrically local systems. Physically, their construction resembles the dissipative dynamics arising from weak system-bath interaction. In this work, we give an efficiently implementable discrete-time counterpart to any continuous-time quantum Gibbs sampler. Our construction preserves the desirable features (i)-(iii) while does not decrease the spectral gap. Also, we give an alternative highly coherent quantum generalization of detailed balanced dynamics that resembles another physically derived master equation, and propose a smooth interpolation between this and earlier constructions. Moreover, we show how to make earlier Metropolis-style Gibbs samplers (which estimate energies both before and after jumps) exactly detailed balanced. We study generic properties of all constructions, including the uniqueness of the fixed point, the (quasi-)locality of the resulting operators. Finally, we prove that the spectral gap of our new highly coherent Gibbs sampler is constant at high temperatures, thereby it mixes fast. We hope that our systematic approach to quantum Glauber and Metropolis dynamics will lead to widespread applications in various domains.
Paper Structure (53 sections, 56 theorems, 313 equations, 2 tables)

This paper contains 53 sections, 56 theorems, 313 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Unifying perspective on classical and quantum detailed balanced dynamics
  3. The new discrete-time constructions
  4. An alternative recursive construction
  5. The spectral gap of the discrete-time quantum channel is always larger
  6. A fully coherent reweighing scheme for achieving exact detailed balance
  7. Mixing times
  8. Structure of the paper
  9. Concurrent work
  10. Discussion and open problems
  11. Classical Metropolis sampling and Glauber dynamics
  12. Discrete-time Metropolis sampling and Glauber dynamics
  13. Continuous-time Metropolis and Glauber dynamics
  14. Quantum dynamics and detailed balance
  15. Quantum channels and Lindbladians
  16. ...and 38 more sections

Key Result

Theorem 1

Consider any $\bm{ \rho}$-detailed balanced CP map $\mathcal{T}'$ such that $\mathcal{T}'^\dagger[\bm{I}] \preceq \bm{I}$. Then, the following CP map with is $\bm{ \rho}$-detailed balanced and trace-preserving.

Theorems & Definitions (105)

  • Theorem 1: Prescription for the rejection part
  • Theorem 2: Time integral representation and LCU implementation
  • Proposition 1: QSVT implementation
  • Theorem 3: Recursively prescribing the reject term
  • Proposition 2
  • Theorem 4
  • proof
  • Theorem 5: Approximating the superoperator $\CS$
  • Theorem 6
  • Theorem 7
  • ...and 95 more