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Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians

Joel Rajakumar, James D. Watson

Abstract

Sampling from Gibbs states -- states corresponding to system in thermal equilibrium -- has recently been shown to be a task for which quantum computers are expected to achieve super-polynomial speed-up compared to classical computers, provided the locality of the Hamiltonian increases with the system size (Bergamaschi et al., arXiv: 2404.14639). We extend these results to show that this quantum advantage still occurs for Gibbs states of Hamiltonians with O(1)-local interactions at constant temperature by showing classical hardness-of-sampling and demonstrating such Gibbs states can be prepared efficiently using a quantum computer. In particular, we show hardness-of-sampling is maintained even for 5-local Hamiltonians on a 3D lattice. We additionally show that the hardness-of-sampling is robust when we are only able to make imperfect measurements.

Gibbs Sampling gives Quantum Advantage at Constant Temperatures with O(1)-Local Hamiltonians

Abstract

Sampling from Gibbs states -- states corresponding to system in thermal equilibrium -- has recently been shown to be a task for which quantum computers are expected to achieve super-polynomial speed-up compared to classical computers, provided the locality of the Hamiltonian increases with the system size (Bergamaschi et al., arXiv: 2404.14639). We extend these results to show that this quantum advantage still occurs for Gibbs states of Hamiltonians with O(1)-local interactions at constant temperature by showing classical hardness-of-sampling and demonstrating such Gibbs states can be prepared efficiently using a quantum computer. In particular, we show hardness-of-sampling is maintained even for 5-local Hamiltonians on a 3D lattice. We additionally show that the hardness-of-sampling is robust when we are only able to make imperfect measurements.
Paper Structure (23 sections, 17 theorems, 46 equations, 1 figure)

This paper contains 23 sections, 17 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Paper Outline
  3. Preliminaries
  4. Notation
  5. Hamiltonian Parameters.
  6. Thermal Physics.
  7. Noisy Circuits.
  8. Parent Hamiltonian Construction for Quantum Circuits
  9. Gibbs States of Parent Hamiltonians
  10. Hardness of Sampling Gibbs States with O(1)-Local Hamiltonians
  11. Overview of Techniques
  12. Hardness of Sampling Gibbs States on 3D Lattices
  13. Hardness of Gibbs Sampling to O(1/poly(N)) Error
  14. Quantum Advantage from Gibbs Sampling of O(1)-Local Hamiltonians
  15. Measurement Errors in Gibbs State Sampling
  16. ...and 8 more sections

Key Result

Theorem 1

There exist two families $\mathcal{F}_1, \mathcal{F}_2$ of efficiently constructable Hamiltonians such that sampling from a probability distribution $Q(x)$ satisfying: is not possible for randomised classical algorithms under complexity-theoretic conjectures, for the following parameter regimes: Sampling from a $Q(x)$ close to either of these distributions can be done efficiently using a quantum

Figures (1)

  • Figure 1: The encoding of the IQP circuit $C$ with $r=3$. For clarity, only two input qubits are shown explicitly, but the procedure applies to all qubits present.

Theorems & Definitions (28)

  • Theorem 1: (Informal) Classically Intractable Gibbs Sampling
  • Definition 2: Circuit Sampling Distributions
  • Lemma 3
  • Lemma 4: Efficient Gibbs State Preparation for Parent Hamiltonians, Lemma 1.2 of bergamaschi2024sample
  • Lemma 5
  • Theorem 6: Classical Hardness of Gibbs Sampling
  • proof
  • Lemma 7
  • Lemma 8
  • Theorem 9
  • ...and 18 more