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Nature Communications

Solving perfect matchings by frequency-grouped multi-photon events using a silicon chip

Pingyu Zhu, Qilin Zheng, Kun Wang, Miaomiao Yu, Gongyu Xia, Jiacheng Liu, Yong Liu, Zhihong Zhu, Ping Xu

Published Apr 22, 2025• 2 citations
Publisher

Abstract

Abstract Computing the number of perfect matchings of a graph is a famous #P-complete problem. In this work, taking the outstanding advantages of the frequency dimension of photon, we propose and implement a photonic perfect matching solver, by combining two key techniques, frequency grouping and multi-photon counting. Based on a broadband photon-pair source from a silicon quantum chip and a wavelength-selective switch, we configured graphs up to sixteen vertices and calculated the perfect matchings of subgraphs up to six vertices. The experimental fidelities are more than 90% for all the graphs. The developed photonic system can be applied in solving nondeterministic-polynomial-time(NP)-complete problems, and we experimentally demonstrate the samples can enhance the random search for solving of the Boolean satisfiability problem and the finding of densest subgraph. Our perfect matching solver avoids destructive interference and does not need to consider the indistinguishability of photons. Moreover, it is simple and convenient to transform or scale up the graph by programming a wavelength-selective switch without changing the photonic source. Our work suggests an excellent platform for graph problem solving.

DOI
10.1038/s41467-025-58711-8

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An important challenge for current and near-term quantum devices is finding useful tasks that can be preformed on them. We first show how to efficiently encode a bounded $n\ifmmode\times\else\texttimes\fi{}n$ matrix AAA into a linear optical circuit with 2n2n2n modes. We then apply this encoding to the case where AAA is a matrix containing information about a graph GGG. We show that a photonic quantum processor consisting of single-photon sources, a linear optical circuit encoding AAA, and single-photon detectors can solve a range of graph problems including finding the number of perfect matchings of bipartite graphs, computing permanental polynomials, determining whether two graphs are isomorphic, and the kkk-densest subgraph problem. We also propose preprocessing methods to boost the probabilities of observing the relevant detection events and thus improve performance. Finally we present both numerical simulations and implementations on Quandela's Ascella photonic quantum processor to validate our findings.
An important challenge for current and near-term quantum devices is finding useful tasks that can be preformed on them. We first show how to efficiently encode a boundedn×nn\times nn×nmatrixAAAinto a linear optical circuit with2n2n2nmodes. We then apply this encoding to the case whereAAAis a matrix containing information about a graphGGG. We show that a photonic quantum processor consisting of single-photon sources, a linear optical circuit encodingAAA, and single-photon detectors can solve a range of graph problems including finding the number of perfect matchings of bipartite graphs, computing permanental polynomials, determining whether two graphs are isomorphic, and the k-densest subgraph problem. We also propose pre-processing methods to boost the probabilities of observing the relevant detection events and thus improve performance. Finally we present both numerical simulations and implementations on Quandela's Ascella photonic quantum processor to validate our findings.
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