Efficient Classical Sampling from Gaussian Boson Sampling Distributions on Unweighted Graphs

TL;DR

This paper introduces a polynomial-time Markov chain Monte Carlo method for sampling from Gaussian Boson Sampling distributions on unweighted graphs, demonstrating improved classical algorithms for graph problems.

Contribution

It presents a novel double-loop Glauber dynamics algorithm with theoretical polynomial mixing time guarantees for dense graphs.

Findings

Improved classical algorithms for max-Hafnian and densest k-subgraph problems.

Experimental validation on graphs with 256 vertices showing 10x performance gains.

Theoretical proof of polynomial mixing time for the proposed Markov chain.

Abstract

Gaussian Boson Sampling (GBS) is a promising candidate for demonstrating quantum computational advantage and can be applied to solving graph-related problems. In this work, we propose Markov chain Monte Carlo-based algorithms to sample from GBS distributions on undirected, unweighted graphs. Our main contribution is a double-loop variant of Glauber dynamics, whose stationary distribution matches the GBS distribution. We further prove that it mixes in polynomial time for dense graphs using a refined canonical path argument. Numerically, we conduct experiments on unweighted graphs with 256 vertices, larger than the scales in former GBS experiments as well as classical simulations. In particular, we show that both the single-loop and double-loop Glauber dynamics improve the performance of original random search and simulated annealing algorithms for the max-Hafnian and densest $k$-subgraph problems up to 10$\times$. Overall, our approach offers both theoretical guarantees and practical advantages for efficient classical sampling from GBS distributions on unweighted graphs.