Performance of Gaussian Boson Sampling on Planted Bipartite Clique Detection

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.