On Quantum Circuits for Discrete Graphical Models

TL;DR

This paper introduces a quantum circuit method for unbiased, independent sampling from discrete graphical models, compatible with current quantum hardware and capable of handling high-dimensional, multi-body interactions.

Contribution

It presents the first provable quantum approach for sampling from discrete factor models, including a novel embedding and a unitary Hammersley-Clifford theorem.

Findings

Method successfully generates unbiased samples on quantum hardware.

Quantum embedding enables maximum likelihood learning and MAP state approximation.

Experiments demonstrate practical implementation on current quantum processors.

Abstract

Graphical models are useful tools for describing structured high-dimensional probability distributions. Development of efficient algorithms for generating unbiased and independent samples from graphical models remains an active research topic. Sampling from graphical models that describe the statistics of discrete variables is a particularly challenging problem, which is intractable in the presence of high dimensions. In this work, we provide the first method that allows one to provably generate unbiased and independent samples from general discrete factor models with a quantum circuit. Our method is compatible with multi-body interactions and its success probability does not depend on the number of variables. To this end, we identify a novel embedding of the graphical model into unitary operators and provide rigorous guarantees on the resulting quantum state. Moreover, we prove a unitary Hammersley-Clifford theorem -- showing that our quantum embedding factorizes over the cliques of the underlying conditional independence structure. Importantly, the quantum embedding allows for maximum likelihood learning as well as maximum a posteriori state approximation via state-of-the-art hybrid quantum-classical methods. Finally, the proposed quantum method can be implemented on current quantum processors. Experiments with quantum simulation as well as actual quantum hardware show that our method can carry out sampling and parameter learning on quantum computers.