Quantum Markov chain Monte Carlo with programmable quantum simulators

TL;DR

The paper addresses the challenge of ergodic sampling over quantum states on near-term hardware by introducing a quantum Markov chain Monte Carlo method that exploits many-body localization (MBL). Moves are generated by Floquet-MBL unitaries in a disordered 1D Ising chain, with a classical Metropolis step determining acceptance, enabling sampling from Gibbs-like or other positive-definite observables. A key theoretical contribution is showing that products of MBL unitaries approach the Haar measure on unitaries (CUE), ensuring irreducibility and enabling a controllable Markov chain; the disorder strength $W$ tunably adjusts move length and the acceptance rate. Practically, the algorithm is demonstrated on MIS, Max-Cut, and HUBO-based prime factorization, highlighting the ability to sample from high-order Hamiltonians and to operate on analog quantum simulators, with extensions to improved convergence and potential 2D implementations and hybrid modeling suggested for future work.

Abstract

In this work, we present a quantum Markov chain algorithm for many-body systems that utilizes a special phase of matter known as the Many-Body Localized (MBL) phase. We show how the properties of the MBL phase enable one to address the conditions for ergodicity and sampling from distributions of quantum states. We demonstrate how to exploit the thermalized-to-localized transition to tune the acceptance rate of the Markov chain, and apply the algorithm to solve a range of combinatorial optimization problems of quadratic order and higher. The algorithm can be implemented on any QPU capable of simulating the Floquet dynamics of a 1D Ising chain with nearest-neighbor interactions, providing a practical way of sampling from thermal distributions of Hamiltonians that cannot be natively implemented on the quantum hardware.

Figures (7)

• Figure 1: The figure illustrates the schematics of the algorithm. The algorithm iteratively proposes new quantum states starting from an initial state $\ket{\psi_0}$. At the $i$-th iteration, a new state $\ket{\psi_i'}$ is proposed by evolving $\ket{\psi_{i-1}}$ with a unitary $U_i'$. The move is evaluated in a Metropolis accept/reject scheme. If accepted $U_i=U_i'$ and $\ket{\psi_i}=\ket{\psi_i'}$, if rejected $U_i=I$ and $\ket{\psi_i}=\ket{\psi_{i-1}}$. Green dots and solid arrows represent accepted moves, while red dots and dashed arrows represent rejected moves. Each move is generated by a unitary operator in the many-body localized (MBL) phase, which allows a controlled exploration of the Hilbert space.
• Figure 2: Numerical simulation of the thermalization of a system of 9 qubits under the MIS cost function for an Erdős-Rényi graph with 0.7 edge probability. (a) The expectation value of the cost function is evaluated at each iteration for Markov chains with varying values of the disorder parameter $W$, and the acceptance rate (A.R.) is calculated. (b) A histogram of bitstring probability against cost found by sampling the state after 50 (blue) and 6000 (orange) iterations of the Markov chain, corresponding to the dotted vertical lines in (a). The disorder parameter is fixed to $W/J=200$. The characteristic exponential behavior of the sampled costs after 6000 iterations indicates the correct thermalization of the Markov chain.
• Figure 3: Probability of observing any of the optimal solutions to (a) MIS and (b) Max-Cut on Erdős-Rényi graphs with 0.5 edge probability, assuming $10^4$ samples of the quantum state per iteration. Four different sets of points are shown, corresponding to stopping the Markov chain at length 100, 150, 200, and 2000. The acceptance rate for MIS ranges between 40% (size 10) and 7% (size 40). For Max-Cut, between 50% (size 10) and 23% (size 30).
• Figure 4: Probability of observing the optimal solution to the integer factorization problem. Each point on the x-axis represents an integer, $M$, for which an independent Markov chain is performed in order to factorize it. The prime factors are encoded with 5 bits for the group of integers on the left (diamond markers) and with 6 bits for the group of integers on the right (square markers). The plot shows the probability of observing the two correct prime factors for varying Markov chain lengths, assuming the state is measured $10^4$ times per iteration. The horizontal lines show the average probability for each Markov chain length. The acceptance rate is around 1% for all instances.
• Figure 5: Convergence of products of MBL unitaries to CUE statistics. The number of matrices in the product is denoted by $M$, and the resulting unitary is denoted $\mathcal{U}_M$. (a) For a system of 5 qubits, histograms of the level spacing of $\mathcal{U}_M$ are plotted for various values of $M=1,\ldots,250$. For $M=1$, the system shows Poissonian statistics typical of MBL (dashed blue line). As $M$ increases, the system quickly converges to CUE statistics (solid blue line). All histograms are generated by collecting the $2^5=32$ eigenvalues of $\mathcal{U}_M$, in an ensemble of 8000 propagators. (b) The Jensen-Shannon distance between the level spacing statistics $Pr(r)$ and CUE as a function of $M$, for systems of 3 to 10 qubits. The distance settles around a value of order $10^{-2}$ for all sizes, with larger systems converging more rapidly.