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Efficient Classical Sampling from Gaussian Boson Sampling Distributions on Unweighted Graphs

Yexin Zhang, Shuo Zhou, Xinzhao Wang, Ziruo Wang, Ziyi Yang, Rui Yang, Yecheng Xue, Tongyang Li

TL;DR

The paper tackles efficient classical sampling from Gaussian Boson Sampling (GBS) distributions on unweighted graphs by developing double-loop Glauber dynamics that yield a stationary distribution proportional to $c^{2|S|} ext{Haf}^2(S)$. It proves polynomial-time mixing for dense graphs via a refined canonical-path framework and inner uniform perfect-matching sampling, and demonstrates practical gains on graph problems such as max-Hafnian and densest $k$-subgraph through extensive simulations on 256-vertex graphs. The approach bridges GBS-inspired sampling with tractable classical algorithms, offering provable guarantees and significant empirical speedups (up to 10× in some settings) over baseline methods. This work provides a concrete, scalable classical tool for benchmarking and solving Hafnian-based graph problems in a regime where GBS is advantageous, while outlining clear avenues for extending guarantees to non-dense and weighted graphs.

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.

Efficient Classical Sampling from Gaussian Boson Sampling Distributions on Unweighted Graphs

TL;DR

The paper tackles efficient classical sampling from Gaussian Boson Sampling (GBS) distributions on unweighted graphs by developing double-loop Glauber dynamics that yield a stationary distribution proportional to . It proves polynomial-time mixing for dense graphs via a refined canonical-path framework and inner uniform perfect-matching sampling, and demonstrates practical gains on graph problems such as max-Hafnian and densest -subgraph through extensive simulations on 256-vertex graphs. The approach bridges GBS-inspired sampling with tractable classical algorithms, offering provable guarantees and significant empirical speedups (up to 10× in some settings) over baseline methods. This work provides a concrete, scalable classical tool for benchmarking and solving Hafnian-based graph problems in a regime where GBS is advantageous, while outlining clear avenues for extending guarantees to non-dense and weighted graphs.

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 -subgraph problems up to 10. Overall, our approach offers both theoretical guarantees and practical advantages for efficient classical sampling from GBS distributions on unweighted graphs.
Paper Structure (25 sections, 28 theorems, 93 equations, 7 figures, 8 algorithms)

This paper contains 25 sections, 28 theorems, 93 equations, 7 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Results
  3. Gaussian Boson Sampling for graph problems
  4. Glauber dynamics for matching
  5. Double-loop Glauber dynamics
  6. Numerical experiments
  7. Glauber dynamics verification and comparison
  8. Score advantage comparison with GBS
  9. Double-loop Glauber dynamics on bipartite graphs
  10. Sparse graph
  11. Discussion
  12. Canonical Path Method
  13. Uniform Sampling for Perfect Matching
  14. Uniform Sampling for Perfect Matchings on Dense Graphs
  15. Uniform Sampling of Perfect Matchings on Bipartite Graphs
  16. ...and 10 more sections

Key Result

Theorem 1

For a general graph $G$ with $n$ vertices and $m$ edges, the mixing time of the Glauber dynamics for the monomer-dimer model on $G$ with fugacity $\lambda>0$ is $O(n^2 m \log n)$.

Figures (7)

  • Figure 1: Flowchart of the Double-loop Glauber Dynamics.
  • Figure 1: A possible canonical path between matchings $I$ and $F$.
  • Figure 2: Glauber Dynamics Verification and Comparison. Iterations $=1000$, mixing time $\leq 10000$, fugacity $\lambda=c^2$, annealing parameter $\gamma=0.95$. (a) Hafnian Random Search on $G_1$ with $k=16$, $c=0.1$. The three enhanced variants are on average $240-300\%$ higher than the original classical algorithm, and the upper confidential intervals are about 350% better. (b) Hafnian Simulated Annealing on $G_1$ with $k=16$, $c=0.1$. The three enhanced variants are on average $20-50\%$ higher than the original classical algorithm, and the upper confidential intervals are about 55% better. (c) Density Random Search on $G_2$ with $k=80$, $c=0.4$. The two enhanced variants are on average $30-35\%$ higher than the original classical algorithm, and the upper confidential intervals are about 32% better. (d) Density Simulated Annealing on $G_2$ with $k=80$, $c=0.4$. The two enhanced variants are on average $10-15\%$ higher than the original classical algorithm, and the upper confidential intervals are about 12% better.
  • Figure 2: A hard instance for non-dense graphs and its two different types of perfect matchings. The graph in the left figure consists of several squares arranged in a cycle, where the first vertex of each square is connected to its third vertex with an edge, and the second vertex of each square is connected to the fourth vertex of the previous square by an edge. The red edges in the right figure form a perfect matching $M_0$ in set $A$ (which is also the unique one); the blue edges form a perfect matching $M_1$ in set $B$. Notice that deleting any edge in $M_0$ will result in a near perfect matching with Hafnian $1$.
  • Figure 3: Score advantage vs. click number: RS enhanced by \ref{['algo:double-loop']} / original RS. Iterations $=100$, mixing time $\leq1000$, fugacity $\lambda=c^2$. (a) The left side reflects the Hafnian score advantage up to 4$\times$ on $G_3$ with $c=0.6$. (b) The right side reflects the density score advantage up to 120% on $G_3$ with $c=0.6$.
  • ...and 2 more figures

Theorems & Definitions (41)

  • Theorem 1: jerrum2003counting
  • Theorem 2
  • Theorem 3
  • Lemma 1: jerrum2004polynomial
  • Lemma 2: jerrum1989approximating
  • Theorem 4
  • Theorem 5
  • Theorem S1: jerrum2003counting
  • Definition S1: Multiple canonical path framework
  • Lemma S3
  • ...and 31 more