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Quantized Markov Chain Couplings that Prepare Qsamples

Kristan Temme, Pawel Wocjan

TL;DR

The paper presents a framework to quantize classical Markov chains by converting couplings into completely positive, trace-preserving quantum maps whose unique fixed point is the qsample $|\sqrt{\pi}\rangle$. The convergence rate of the quantum dynamics is tightly linked to the classical coupling time, offering a direct route to preparing qsamples from arbitrary starting states. It provides both theoretical bounds and practical schemes for implementing quantized grand couplings, including concrete examples (random walk on the hypercube, random colorings, hardcore model) and a general theorem on efficient implementation for sparse couplings. This approach offers an alternative to Szegedy-based quantizations and has potential implications for quantum algorithms in partition function estimation, Bayesian inference, and sampling tasks. Open problems include extending CP-guarantees to broader classes of couplings and refining convergence constants.

Abstract

We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling.

Quantized Markov Chain Couplings that Prepare Qsamples

TL;DR

The paper presents a framework to quantize classical Markov chains by converting couplings into completely positive, trace-preserving quantum maps whose unique fixed point is the qsample . The convergence rate of the quantum dynamics is tightly linked to the classical coupling time, offering a direct route to preparing qsamples from arbitrary starting states. It provides both theoretical bounds and practical schemes for implementing quantized grand couplings, including concrete examples (random walk on the hypercube, random colorings, hardcore model) and a general theorem on efficient implementation for sparse couplings. This approach offers an alternative to Szegedy-based quantizations and has potential implications for quantum algorithms in partition function estimation, Bayesian inference, and sampling tasks. Open problems include extending CP-guarantees to broader classes of couplings and refining convergence constants.

Abstract

We present a novel approach to quantizing Markov chains. The approach is based on the Markov chain coupling method, which is frequently used to prove fast mixing. Given a particular coupling, e.g., a grand coupling, we construct a completely positive and trace preserving map. This quantum map has a unique fixed point, which corresponds to the quantum sample (qsample) of the classical Markov chain's stationary distribution. We show that the convergence time of the quantum map is directly related to the coupling time of the Markov chain coupling.

Paper Structure

This paper contains 13 sections, 16 theorems, 97 equations.

Table of Contents

  1. Introduction
  2. Markov chain couplings
  3. Examples of couplings
  4. Quantized Markov chain couplings
  5. Completely positive quantized couplings
  6. Convergence rate of quantized Markov chain couplings
  7. Implementation of quantized grand couplings
  8. Simulation of sparse tcp-maps
  9. Random walk on the hypercube
  10. Random colorings
  11. Hardcore model with fugacity
  12. General theorem
  13. Conclusions and open problems

Key Result

Theorem 1

Assume we are given a Markov chain coupling for an ergodic Markov chain $P$ with stationary distribution $\pi=(\pi_x : x \in \Omega)$. We construct a corresponding trace preserving map $\mathcal{T}$ called quantized coupling. For independent and grand couplings, the resulting quantized coupling $\ma whenever where $\rho$ is any initial state, $\pi_* = \min\{ \pi_x : x \in \Omega\}$. The quantized

Theorems & Definitions (33)

  • Theorem 1: main results (informal)
  • Theorem 2: Connection between coalescence and mixing times
  • Remark 3
  • Definition 4: Independent coupling
  • Definition 5: Grand coupling
  • Definition 6: Quantized coupling
  • Lemma 7: Trace preserving
  • proof
  • Lemma 8: Fixed point
  • proof
  • ...and 23 more