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Quantum Circuits for the Metropolis-Hastings Algorithm

Baptiste Claudon, Pablo Rodenas-Ruiz, Jean-Philip Piquemal, Pierre Monmarché

TL;DR

The paper tackles speeding Metropolis-Hastings simulations on quantum hardware by grafting Szegedy's quantum walk onto a dual MH kernel that operates on edges rather than states. It develops explicit projected-unitary encodings and circuits that realize the dual proposal and acceptance steps with a constant ancilla footprint, avoiding heavy reversible-computation overhead. A key theoretical result is that the qubitized walk exhibits a spectral- (angular-) gap of $\Omega\big(\sqrt{\delta}\big)$, preserving the quadratic speedup in end-to-end MH sampling; in the Glauber case the gap matches the classical one, and for general acceptance matrices a $\delta/2$ bound holds with a lazy MH adjustment to ensure ergodicity. The authors demonstrate the construction on Metropolis-Adjusted Langevin dynamics (MALA) with a scalable 4m+3 qubit circuit and provide open-source code, highlighting practical feasibility on near-term fault-tolerant devices.

Abstract

Szegedy's quantization of a reversible Markov chain provides a quantum walk whose mixing time is quadratically smaller than that of the classical walk. Quantum computers are therefore expected to provide a speedup of Metropolis-Hastings (MH) simulations. Existing generic methods to implement the quantum walk require coherently computing the acceptance probabilities of the underlying Markov kernel. However, reversible computing methods require a number of qubits that scales with the complexity of the computation. This overhead is undesirable in near-term fault-tolerant quantum computing, where few logical qubits are available. In this work, we present a quantum walk construction which follows the classical proposal-acceptance logic, does not require further reversible computing methods, and uses a constant-sized ancilla register. Since each step of the quantum walk uses a constant number of proposition and acceptance steps, we expect the end-to-end quadratic speedup to hold for MH Markov Chain Monte-Carlo simulations.

Quantum Circuits for the Metropolis-Hastings Algorithm

TL;DR

The paper tackles speeding Metropolis-Hastings simulations on quantum hardware by grafting Szegedy's quantum walk onto a dual MH kernel that operates on edges rather than states. It develops explicit projected-unitary encodings and circuits that realize the dual proposal and acceptance steps with a constant ancilla footprint, avoiding heavy reversible-computation overhead. A key theoretical result is that the qubitized walk exhibits a spectral- (angular-) gap of , preserving the quadratic speedup in end-to-end MH sampling; in the Glauber case the gap matches the classical one, and for general acceptance matrices a bound holds with a lazy MH adjustment to ensure ergodicity. The authors demonstrate the construction on Metropolis-Adjusted Langevin dynamics (MALA) with a scalable 4m+3 qubit circuit and provide open-source code, highlighting practical feasibility on near-term fault-tolerant devices.

Abstract

Szegedy's quantization of a reversible Markov chain provides a quantum walk whose mixing time is quadratically smaller than that of the classical walk. Quantum computers are therefore expected to provide a speedup of Metropolis-Hastings (MH) simulations. Existing generic methods to implement the quantum walk require coherently computing the acceptance probabilities of the underlying Markov kernel. However, reversible computing methods require a number of qubits that scales with the complexity of the computation. This overhead is undesirable in near-term fault-tolerant quantum computing, where few logical qubits are available. In this work, we present a quantum walk construction which follows the classical proposal-acceptance logic, does not require further reversible computing methods, and uses a constant-sized ancilla register. Since each step of the quantum walk uses a constant number of proposition and acceptance steps, we expect the end-to-end quadratic speedup to hold for MH Markov Chain Monte-Carlo simulations.

Paper Structure

This paper contains 17 sections, 24 theorems, 81 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Use case: Metropolis-Adjusted Langevin Algorithm
  3. Technical background
  4. Ergodic Markov kernels and their mixing time
  5. Reversible Markov chains
  6. Projected unitary encodings and qubitized walk operators
  7. Szegedy quantization of Markov chains
  8. Metropolis-Hastings algorithm
  9. Results
  10. Problem statement
  11. Dual Metropolis-Hastings kernel
  12. Dual proposal $\mathcal{T}$ and acceptance $\mathcal{A}$ kernels
  13. Quantum circuits for the dual quantum walk
  14. Gap of the qubitized walk operator
  15. The Glauber case
  16. ...and 2 more sections

Key Result

Proposition 1

Let $P$ be a Markov chain with a single non-degenerate eigenvalue on the unit circle. Then, there exists a function $\pi:\mathbb S\to [0,1]$ such that:

Figures (5)

  • Figure 1: Spectral properties of the implemented walk operator $\mathcal{W}$. The process under study is the MALA to target a measure $\pi\propto \exp(-U$ for a two-well potential $U$. $\delta$ is the spectral gap of the classical walk. $\mathcal{W}$ presents a quadratically amplified spectral gap.
  • Figure 2: Graphical representation of a MH step. Step 1. The process is in the state $x$, with neighbours $y, z$. Step 2. Select the outgoing edge $(x, y)$ with probability $T(x, y)$ and the outgoing edge $(x, z)$ with probability $T(x, z)$. Step 3. Given the current edge $(a, b)$, let the process be in state $b$ with probability $A(a, b)$ and be in state $a$ with probability $1-A(a, b)$.
  • Figure 3: Graphical representation of a $\mathcal{TAT}$ step. Step 1. The process is in the state $(x, y)$. Step 2. Select the new head $y$ with probability $T(x, y)$ and the new head $z$ with probability $T(x, z)$. Step 3. Given the current edge $(a, b)$, flip it to $(b, a)$ with probability $A(a, b)$. Step 4. Sample a new head for the current edge as in Step 1.
  • Figure 4: Quantum circuit for $O_{\mathcal{A}}$.
  • Figure 5: Quantum circuit for the final qubitized walk operator $\mathcal{W}$. Recall that we are quantizing a walk on the edges $\mathcal{S}$ of the state space so that the unitaries $O, O_\star$ act on a total of 4 registers of size $m=\log|\mathbb S|$.

Theorems & Definitions (60)

  • Definition 1
  • Definition 2
  • Proposition 1
  • Definition 3
  • Definition 4
  • Proposition 2
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • ...and 50 more