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Optimal quantum algorithm for Gibbs state preparation

Cambyse Rouzé, Daniel Stilck França, Álvaro M. Alhambra

TL;DR

<3-5 sentence high-level summary> The paper addresses the challenge of efficiently thermalizing open quantum many-body systems by proposing a dissipative quantum Gibbs sampler whose generator is simulable on quantum hardware. It proves rapid mixing to the Gibbs state at sufficiently high temperature for both local and long-range Hamiltonians, using the oscillator norm, Lindblad approximations, and Lieb-Robinson bounds to obtain explicit high-temperature thresholds. This rapid mixing enables quasi-linear quantum-time preparation of Gibbs states and underpins a quantum algorithm to estimate quantum partition functions with polynomial speedups over classical methods for short-range systems and the first provably efficient approach for long-range interactions. The results significantly advance the understanding of high-temperature quantum Gibbs samplers and open pathways for efficient quantum thermodynamics computations on near- and mid-term quantum devices.

Abstract

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced disispative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then employ our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

Optimal quantum algorithm for Gibbs state preparation

TL;DR

<3-5 sentence high-level summary> The paper addresses the challenge of efficiently thermalizing open quantum many-body systems by proposing a dissipative quantum Gibbs sampler whose generator is simulable on quantum hardware. It proves rapid mixing to the Gibbs state at sufficiently high temperature for both local and long-range Hamiltonians, using the oscillator norm, Lindblad approximations, and Lieb-Robinson bounds to obtain explicit high-temperature thresholds. This rapid mixing enables quasi-linear quantum-time preparation of Gibbs states and underpins a quantum algorithm to estimate quantum partition functions with polynomial speedups over classical methods for short-range systems and the first provably efficient approach for long-range interactions. The results significantly advance the understanding of high-temperature quantum Gibbs samplers and open pathways for efficient quantum thermodynamics computations on near- and mid-term quantum devices.

Abstract

It is of great interest to understand the thermalization of open quantum many-body systems, and how quantum computers are able to efficiently simulate that process. A recently introduced disispative evolution, inspired by existing models of open system thermalization, has been shown to be efficiently implementable on a quantum computer. Here, we prove that, at high enough temperatures, this evolution reaches the Gibbs state in time scaling logarithmically with system size. The result holds for Hamiltonians that satisfy the Lieb-Robinson bound, such as local Hamiltonians on a lattice, and includes long-range systems. To the best of our knowledge, these are the first results rigorously establishing the rapid mixing property of high-temperature quantum Gibbs samplers, which is known to give the fastest possible speed for thermalization in the many-body setting. We then employ our result to the problem of estimating partition functions at high temperature, showing an improved performance over previous classical and quantum algorithms.

Paper Structure

This paper contains 19 sections, 9 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Quantum Gibbs sampling
  3. Rapid mixing for high-temperature quantum Gibbs sampler
  4. Local Hamiltonians
  5. The oscillator norm as a proxy:
  6. Local conditions for oscillator norm decay:
  7. Local conditions via Lindblad approximations:
  8. Spatial and thermal approximations via Lieb-Robinson bounds:
  9. Long-range Hamiltonians
  10. Approximating quantum partition functions
  11. Conclusion
  12. Rapid mixing at high temperature
  13. Local conditions for oscillator norm decay
  14. Local conditions via Lindblad approximations
  15. Locality analysis
  16. ...and 4 more sections

Key Result

Theorem 1

Let $H$ be $(k,l)$-local on a $D$ dimensional lattice, and let $\beta^*=\frac{1}{615^{D}J}$. Then, for any $\beta<\beta^*$ and any initial state $\rho$,

Theorems & Definitions (12)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Lemma 4
  • Lemma 5: Block encoding of the exponential function vanApeldoorn2020
  • Lemma 6
  • proof
  • Corollary 7
  • proof
  • ...and 2 more