Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Code Swendsen-Wang Dynamics

Dominik Hangleiter, Nathan Ju, Umesh Vazirani

TL;DR

The work tackles the challenge of efficiently preparing quantum Gibbs states for commuting Hamiltonians, especially when energy barriers hinder local dynamics at low temperatures. It introduces Code SW dynamics, a quantum Markov chain that leverages a classical lifting from code-based Gibbs sampling to achieve rapid mixing for codes with graphic or near-graphic parity checks, including the 4D toric code, at all temperatures. The authors prove polynomial-time mixing for $\Delta$-graphic or $\Delta$-cographic codes by constructing good flows via primal/dual couplings and worm-based canonical paths, and they connect quantum Gibbs sampling to classical RC and even-cover models. This provides a discrete, scalable route to efficient Gibbs-state preparation across a broad class of quantum codes and suggests near-term applicability to stabilizer codes beyond the 2D/4D toric instances, with the caveat of first-order phase-transition points. The approach has significant impact on quantum simulation and error-correcting code design by enabling rapid convergence to Gibbs states from arbitrary initial configurations.

Abstract

An important open question about Markov chains for preparing quantum Gibbs states is proving rapid mixing. However, rapid mixing at low temperatures has only been proven for Gibbs states with no thermally stable phases, e.g., the 2D toric code. Inspired by Swendsen-Wang dynamics, in this work we give a simple Markov chain, Code Swendsen-Wang dynamics, for preparing Gibbs states of commuting Hamiltonians. We prove rapid mixing of this chain for classes of quantum and classical Hamiltonians with thermally stable phases, including the 4D toric code, at any temperature. We conjecture its efficiency for all code Hamiltonians away from first-order phase transition points.

Code Swendsen-Wang Dynamics

TL;DR

The work tackles the challenge of efficiently preparing quantum Gibbs states for commuting Hamiltonians, especially when energy barriers hinder local dynamics at low temperatures. It introduces Code SW dynamics, a quantum Markov chain that leverages a classical lifting from code-based Gibbs sampling to achieve rapid mixing for codes with graphic or near-graphic parity checks, including the 4D toric code, at all temperatures. The authors prove polynomial-time mixing for -graphic or -cographic codes by constructing good flows via primal/dual couplings and worm-based canonical paths, and they connect quantum Gibbs sampling to classical RC and even-cover models. This provides a discrete, scalable route to efficient Gibbs-state preparation across a broad class of quantum codes and suggests near-term applicability to stabilizer codes beyond the 2D/4D toric instances, with the caveat of first-order phase-transition points. The approach has significant impact on quantum simulation and error-correcting code design by enabling rapid convergence to Gibbs states from arbitrary initial configurations.

Abstract

An important open question about Markov chains for preparing quantum Gibbs states is proving rapid mixing. However, rapid mixing at low temperatures has only been proven for Gibbs states with no thermally stable phases, e.g., the 2D toric code. Inspired by Swendsen-Wang dynamics, in this work we give a simple Markov chain, Code Swendsen-Wang dynamics, for preparing Gibbs states of commuting Hamiltonians. We prove rapid mixing of this chain for classes of quantum and classical Hamiltonians with thermally stable phases, including the 4D toric code, at any temperature. We conjecture its efficiency for all code Hamiltonians away from first-order phase transition points.

Paper Structure

This paper contains 22 sections, 16 theorems, 89 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. (Classical) code SW dynamics
  3. Discussion and related work
  4. Technical overview
  5. Quantum Markov chain for code Hamiltonians
  6. Preliminaries: Codes and their Gibbs states
  7. Linear codes and stabilizer codes
  8. Gibbs states of quantum and classical codes
  9. Lifting classical code sampling to quantum code sampling
  10. Quantum Markov chains beyond code Hamiltonians
  11. Code Swendsen-Wang dynamics and mixing time
  12. Recap: Swendsen-Wang dynamics for the Ising model
  13. Code Swendsen-Wang dynamics
  14. Mixing time of the code Swendsen-Wang dynamics
  15. Comparison to single-check updates
  16. ...and 7 more sections

Key Result

Theorem 1

The code SW algorithm for the 4D toric code mixes rapidly at any temperature.

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2: Rapid mixing for $\Delta$-graphic or $\Delta$-cographic codes
  • Theorem 3: Convergence to the Gibbs state
  • proof
  • Lemma 4: Coupling of quantum and classical chains
  • proof
  • Claim 5: Coupling of Ising and RC model
  • proof
  • Lemma 6: Correctness of code SW dynamics
  • proof
  • ...and 27 more