Gaussian Boson Sampling to Accelerate NP-Complete Vertex-Minor Graph Classification

Abstract: Gaussian Boson Sampling (GBS) generate random samples of photon-click patterns from a class of probability distributions that are hard for a classical computer to sample from. Despite heroic demonstrations for quantum supremacy using GBS, Boson Sampling, and instantaneous quantum polynomial (IQP) algorithms, systematic evaluations of the power of these quantum-enhanced random samples when applied to provably hard problems, and performance comparisons with best-known classical algorithms have been lacking. We propose a hybrid quantum-classical algorithm using the GBS for the NP-complete problem of determining if two graphs are vertex minor of one another. The graphs are encoded in GBS and the generated random samples serve as feature vectors in the support vector machine (SVM) classifier. We find a graph embedding that allows trading between the one-shot classification accuracy and the amount of input squeezing, a hard-to-produce quantum resource, followed by repeated trials and majority vote to reach an overall desired accuracy. We introduce a new classical algorithm based on graph spectra, which we show outperforms various well-known graph-similarity algorithms. We compare the performance of our algorithm with this classical algorithm and analyze their time vs problem-size scaling, to yield a desired classification accuracy. Our simulation suggests that with a near-term realizable GBS device- $5$ dB pulsed squeezer, $12$-mode unitary, and reasonable assumptions on coupling efficiency, on-chip losses and detection efficiency of photon number resolving detectors-we can solve a $12$-node vertex minor instances with about $10^3$ fold lower time compared to a powerful desktop computer.