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Performance of Gaussian Boson Sampling on Planted Bipartite Clique Detection

Yu-Zhen Janice Chen, Laurent Massoulié, Don Towsley

TL;DR

This paper investigates whether Gaussian Boson Sampling (GBS) can provide a quantum advantage for detecting planted bicliques in random bipartite graphs. By defining node weights as the GBSer-related frequency of vertex inclusion and analyzing their moments, the authors show that in the classical hard regime where $2\log n \ll K \ll \sqrt{n}$, fluctuations dominate the planted-signal bias, making simple weight-based detection unreliable; a detectable signal emerges only at $K=\Theta(\sqrt{n})$. They establish that the joint weight distribution converges to a multivariate Gaussian and quantify the bias scaling as $\sim (K/n)$, providing rigorous evidence for the difficulty of planted biclique detection under ideal GBS. The work also derives a lognormal limit for the Hafnian of bipartite Erdős–Rényi graphs and discusses broader implications for post-quantum cryptography and future quantum algorithms beyond node-frequency statistics.

Abstract

We investigate whether Gaussian Boson Sampling (GBS) can provide a computational advantage for solving the planted biclique problem, which is a graph problem widely believed to be classically hard when the planted structure is small. Although GBS has been heuristically and experimentally observed to favor sampling dense subgraphs, its theoretical performance on this classically hard problem remains largely unexplored. We focus on a natural statistic derived from GBS output: the frequency with which a node appears in GBS samples, referred to as the node weight. We rigorously analyze whether this signal is strong enough to distinguish planted biclique nodes from background nodes. Our analysis characterizes the distribution of node weights under GBS and quantifies the bias introduced by the planted structure. The results reveal a sharp limitation: when the planted biclique size falls within the conjectured hard regime, the natural fluctuations in node weights dominate the bias signal, making detection unreliable using simple ranking strategies. These findings provide the first rigorous evidence that planted biclique detection may remain computationally hard even under GBS-based quantum computing, and they motivate further investigation into more advanced GBS-based algorithms or other quantum approaches for this problem.

Performance of Gaussian Boson Sampling on Planted Bipartite Clique Detection

TL;DR

This paper investigates whether Gaussian Boson Sampling (GBS) can provide a quantum advantage for detecting planted bicliques in random bipartite graphs. By defining node weights as the GBSer-related frequency of vertex inclusion and analyzing their moments, the authors show that in the classical hard regime where , fluctuations dominate the planted-signal bias, making simple weight-based detection unreliable; a detectable signal emerges only at . They establish that the joint weight distribution converges to a multivariate Gaussian and quantify the bias scaling as , providing rigorous evidence for the difficulty of planted biclique detection under ideal GBS. The work also derives a lognormal limit for the Hafnian of bipartite Erdős–Rényi graphs and discusses broader implications for post-quantum cryptography and future quantum algorithms beyond node-frequency statistics.

Abstract

We investigate whether Gaussian Boson Sampling (GBS) can provide a computational advantage for solving the planted biclique problem, which is a graph problem widely believed to be classically hard when the planted structure is small. Although GBS has been heuristically and experimentally observed to favor sampling dense subgraphs, its theoretical performance on this classically hard problem remains largely unexplored. We focus on a natural statistic derived from GBS output: the frequency with which a node appears in GBS samples, referred to as the node weight. We rigorously analyze whether this signal is strong enough to distinguish planted biclique nodes from background nodes. Our analysis characterizes the distribution of node weights under GBS and quantifies the bias introduced by the planted structure. The results reveal a sharp limitation: when the planted biclique size falls within the conjectured hard regime, the natural fluctuations in node weights dominate the bias signal, making detection unreliable using simple ranking strategies. These findings provide the first rigorous evidence that planted biclique detection may remain computationally hard even under GBS-based quantum computing, and they motivate further investigation into more advanced GBS-based algorithms or other quantum approaches for this problem.
Paper Structure (34 sections, 10 theorems, 154 equations, 1 algorithm)

This paper contains 34 sections, 10 theorems, 154 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background and Problem Setup
  3. Planted Biclique Problem
  4. Gaussian Boson Sampling
  5. Methodology
  6. Definitions of Weights
  7. Key Technical Tools
  8. Characterization of Node weight
  9. Evaluation of Moments
  10. Evaluation of Bias
  11. Consequences on Planted Clique Detection
  12. Discussion
  13. Planted Clique and Post-Quantum Cryptography.
  14. Beyond Node Frequencies in GBS-Based Detection.
  15. Distribution of Normalized Hafnian in Bipartite Erdös-Rényi Graph.
  16. ...and 19 more sections

Key Result

Lemma 1

Let $\mathcal{C} \subseteq \{0,\ldots,\ell\}$ with $|\mathcal{C}| \ge 2$, and define Fix integer values $x_{\mathcal{C}} \in \mathbb{N}$ for all $\mathcal{C}$ with $|\mathcal{C}| \ge 2$, and let Then, the following holds:

Theorems & Definitions (24)

  • Lemma 1: Vertex Subsets Intersection Size
  • Lemma 2: Bijections Intersection Size
  • Theorem 3: Joint Moment of Centered and Rescaled Weights
  • Corollary 4: Weak Convergence to Multivariate Gaussian
  • proof
  • Remark 5
  • Proposition 6
  • Corollary 7
  • proof
  • Remark 8
  • ...and 14 more