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Simulating Gaussian boson sampling on graphs in polynomial time

Konrad Anand, Zongchen Chen, Mary Cryan, Graham Freifeld, Leslie Ann Goldberg, Heng Guo, Xinyuan Zhang

TL;DR

The paper addresses whether Gaussian Boson Sampling on graphs, and a related non-negative Boson Sampling distribution, can be classically simulated in polynomial time. It achieves this by reducing the sampling task to weighted perfect-matchings on the Cartesian product graph $G\square K_2$ and applying the Jerrum–Sinclair chain with a carefully chosen weight scaling, yielding a runtime of $O(\overline{c} m n^4 \log^2(n\overline{c}/\varepsilon))$. For standard Boson Sampling with non-negative input matrices, it provides a similar polynomial-time sampler with runtime $O\left(\frac{m^7 n^{14}}{\varepsilon^7} \log^4\left(\frac{mn}{\varepsilon}\right)\right)$. Together, these results challenge the expectation of exponential quantum advantage for graph-based applications of BS/GBS in the non-negative regime and motivate further exploration of relaxing assumptions and linking to permanents in more general settings.

Abstract

We show that a distribution related to Gaussian Boson Sampling (GBS) on graphs can be sampled classically in polynomial time. Graphical applications of GBS typically sample from this distribution, and thus quantum algorithms do not provide exponential speedup for these applications. We also show that another distribution related to Boson sampling can be sampled classically in polynomial time.

Simulating Gaussian boson sampling on graphs in polynomial time

TL;DR

The paper addresses whether Gaussian Boson Sampling on graphs, and a related non-negative Boson Sampling distribution, can be classically simulated in polynomial time. It achieves this by reducing the sampling task to weighted perfect-matchings on the Cartesian product graph and applying the Jerrum–Sinclair chain with a carefully chosen weight scaling, yielding a runtime of . For standard Boson Sampling with non-negative input matrices, it provides a similar polynomial-time sampler with runtime . Together, these results challenge the expectation of exponential quantum advantage for graph-based applications of BS/GBS in the non-negative regime and motivate further exploration of relaxing assumptions and linking to permanents in more general settings.

Abstract

We show that a distribution related to Gaussian Boson Sampling (GBS) on graphs can be sampled classically in polynomial time. Graphical applications of GBS typically sample from this distribution, and thus quantum algorithms do not provide exponential speedup for these applications. We also show that another distribution related to Boson sampling can be sampled classically in polynomial time.

Paper Structure

This paper contains 7 sections, 9 theorems, 21 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Classical algorithm to sample (perfect) matchings
  4. Permanent and Hafnian
  5. Boson sampling
  6. Gaussian boson sampling
  7. Concluding remarks

Key Result

Theorem 1.1

There is an algorithm that, given a graph $G=(V,E)$ and positive real numbers $c$ and $\varepsilon$, samples from a distribution that is $\varepsilon$-close to $\mu_{GBS,G}$ in total variation distance, in time $O(\overline{c}mn^4 \log^2 (n\overline{c}/\varepsilon))$ where $m=\left\vert E\right\vert

Figures (3)

  • Figure 1: On the left we have our graph $G$ with vertex sets from left to right: $L_1, R_1, R_2, L_2$. The corresponding matrix $A$ is a $2\times 3$ matrix with all $1$ entries, and $k=2$ in this example. On the right, a perfect matching $M$ selected from $G$ is highlighted. Here $S_1(M) = \{u_{1,1}^{(1)},u_{2,1}^{(1)}\}$ and $\mathbf{z}=\{1,1\}$.
  • Figure 2: On the left we have our original graph $G$, while on the right we have $G \mathbin{\text{$\square$}} K_2.$
  • Figure 3: On the left, we have a perfect matching $M$ in $G \mathbin{\text{$\square$}} K_2$, with $M \cap E$ highlighted in red in the top copy, $M\cap E'$ blue in the bottom copy, and $M \cap E_0$ black. On the right, we have our underlying graph $G$ with the set $S_M$ and the corresponding edges chosen by $M$ (in either or both copies) highlighted.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Theorem 3.1
  • proof
  • Lemma 4.1
  • proof
  • ...and 4 more