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Mixing time of quantum Gibbs sampling for random sparse Hamiltonians

Akshar Ramkumar, Mehdi Soleimanifar

TL;DR

A polylog(n) upper bound is established on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature for this newly developed quantum Gibbs sampling algorithm.

Abstract

Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm by Chen, Kastoryano, and Gilyén provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.

Mixing time of quantum Gibbs sampling for random sparse Hamiltonians

TL;DR

A polylog(n) upper bound is established on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature for this newly developed quantum Gibbs sampling algorithm.

Abstract

Providing evidence that quantum computers can efficiently prepare low-energy or thermal states of physically relevant interacting quantum systems is a major challenge in quantum information science. A newly developed quantum Gibbs sampling algorithm by Chen, Kastoryano, and Gilyén provides an efficient simulation of the detailed-balanced dissipative dynamics of non-commutative quantum systems. The running time of this algorithm depends on the mixing time of the corresponding quantum Markov chain, which has not been rigorously bounded except in the high-temperature regime. In this work, we establish a polylog(n) upper bound on its mixing time for various families of random n by n sparse Hamiltonians at any constant temperature. We further analyze how the choice of the jump operators for the algorithm and the spectral properties of these sparse Hamiltonians influence the mixing time. Our result places this method for Gibbs sampling on par with other efficient algorithms for preparing low-energy states of quantumly easy Hamiltonians.

Paper Structure

This paper contains 37 sections, 15 theorems, 83 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Our main results
  3. Choice of jumps: graph-local vs unitary design
  4. Mixing time from spectral profile
  5. Explicit examples of random sparse Hamiltonians
  6. Proof sketch
  7. Graph-local jumps
  8. Unitary design jumps
  9. Technical details
  10. The quantum Gibbs sampler
  11. Lindbladian evolution
  12. Detailed balance
  13. Construction and parameters
  14. Spectral gap
  15. Unbounded degree systems
  16. ...and 22 more sections

Key Result

Theorem 2.1

Fix temperature $\beta^{-1}$. There exists some constant energy resolution $\sigma_E$ for which the spectral gap of the CKG Lindbladian $\mathcal{L}_\beta$ for a cyclic graph with $n$ vertices with jump operators $\bm{A}^a = \frac{1}{\sqrt{n}}\ketbra{e_a}{e_a}$ is asymptotically $\Theta(n^{-3})$.

Figures (1)

  • Figure 1: Linear (above) and log-log (below) graphs of spectral gap with respect to system size. Gaps of ten random 4-regular graphs were averaged for each data point. For the cyclic graphs (one-dimensional lattices), the proven asymptotic decay aligns closely with the data.

Theorems & Definitions (24)

  • Theorem 2.1: Spectral gap of cyclic graphs with local jumps
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.2: Constant spectral gap of Lindbladian in bounded degree systems
  • Definition 2.3
  • Theorem 2.3: Spectral gap of Lindbladian in unbounded degree systems
  • Lemma 3.1: Constant spectral gap of average Lindbladian for bounded degree systems
  • Lemma 3.2: Spectral gap of average Lindbladian for unbounded systems
  • Theorem 4.1
  • Theorem 4.2
  • ...and 14 more