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arXiv:2512.24423

A Quantum-Inspired Algorithm for Graph Isomorphism

Innes L. Maxwell, Stefan N. van den Hoven, Jelmer J. Renema

quant-phSubmitted Dec 30, 2025• Announced Jan 1, 2026• 0 citations• Revised Jun 2, 2026 (v3)
v1 · Dec 30, 2025v2 · Jun 1, 2026v3 · Jun 2, 2026
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Abstract

The Noisy Intermediate-Scale Quantum (NISQ) era of technology in which we currently find ourselves is defined by non-universality, susceptibility to errors and noise, and a search for useful applications. While demonstrations of practical quantum advantage remain elusive in this era, it provides space to develop and analyze the advantages and limitations of systems and their ability to solve problems. In this work, we critically assess a proposed quantum algorithm for the graph isomorphism problem, implemented on a photonic quantum device. Inspired by the nature of this quantum algorithm, we formulate a necessary condition for the isomorphism of graphs encoded in Gaussian boson samplers and a classical algorithm to test for it. Our classical algorithm makes use of efficiently computable statistical properties of a quantum sampling system to show a pair of graphs fail to meet our necessary condition and thus cannot be isomorphic. We analyze our algorithm in the context of the inspiring, sampler-based quantum algorithm of Bràdler et. al., the classical color refinement algorithm, and the state-of-the-art quasi-polynomial Babai algorithm.

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The Noisy Intermediate-Scale Quantum (NISQ) era of technology in which we currently find ourselves is defined by non-universality, susceptibility to errors and noise, and a search for useful applications. While demonstrations of practical quantum advantage remain elusive in this era, it provides space to develop and analyze the advantages and limitations of systems and their ability to solve problems. In this work, we critically assess a proposed quantum algorithm for the graph isomorphism problem, implemented on a photonic quantum device. Inspired by the nature of this quantum algorithm, we formulate a necessary condition for the isomorphism of graphs encoded in Gaussian boson samplers and a classical algorithm to test for it. Our classical algorithm makes use of efficiently computable statistical properties of a quantum sampling system to show a pair of graphs fail to meet our necessary condition and thus cannot be isomorphic. We analyze our algorithm in the context of the inspiring, sampler-based quantum algorithm of Bràdler et. al., the classical color refinement algorithm, and the state-of-the-art quasi-polynomial Babai algorithm.
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Gaussian boson sampling (GBS) is not only a feasible protocol for demonstrating quantum computational advantage, but also mathematically associated with certain graph-related and quantum chemistry problems. In particular, it is proposed that the generated samples from the GBS could be harnessed to enhance the classical stochastic algorithms in searching some graph features. Here, we use Jiǔzhāng, a noisy intermediate-scale quantum computer, to solve graph problems. The samples are generated from a 144-mode fully connected photonic processor, with photon click up to 80 in the quantum computational advantage regime. We investigate the open question of whether the GBS enhancement over the classical stochastic algorithms persists-and how it scales-with an increasing system size on noisy quantum devices in the computationally interesting regime. We experimentally observe the presence of GBS enhancement with a large photon-click number and a robustness of the enhancement under certain noise. Our work is a step toward testing real-world problems using the existing noisy intermediate-scale quantum computers and hopes to stimulate the development of more efficient classical and quantum-inspired algorithms.
The noisy intermediate-scale quantum (NISQ) computer significantly advances quantum computing technology. Due to the physical connectivity constraints of the NISQ device, its induced qubit mapping problem becomes more challenging. Recent works employ heuristics to achieve promising outcomes. However, they are limited to using only one type of center for graph matching, and their exhaustive traversal of the coupling graph results in high computation time. This paper strategically generates specific subgraphs during the initial mapping stage to reduce the solution space for the coupling graph. Then, we employ a bidirectional graph isomorphism search to improve initial mapping. In the main mapping stage, we develop an efficient search algorithm to minimize the number of inserted gates. Experimental results show that our method significantly outperforms the state-of-the-art work in reducing the number of inserted CNOT gates by 13.22% and the runtime by 19.61%.
Quantum photonic processors are emerging as promising platforms to prove preliminary evidence of quantum computational advantage toward the realization of universal quantum computers. In the context of nonuniversal noisy intermediate quantum devices, photonic-based sampling machines solving the Gaussian boson sampling (GBS) problem currently play a central role in the experimental demonstration of quantum computational advantage. A relevant issue is the validation of the sampling process in the presence of experimental noise, such as photon losses, which could undermine the hardness of simulating the experiment. We test the capability of a validation protocol that exploits the connection between GBS and graph perfect match counting to perform such an assessment in a noisy scenario. In particular, we use as a test bench the recently developed machine Borealis, a large-scale sampling machine that has been made available online for external users, and address its operation in the presence of noise. The employed approach to validation is also shown to provide connections with the open question on the effective advantage of using noisy GBS devices for graph similarity and isomorphism problems and thus provides an effective method for certification of quantum hardware.
The graph isomorphism problem asks whether two graphs are equivalent up to a permutation of their vertices. Its precise computational complexity remains unresolved, and it is still unknown whether a polynomial-time algorithm exists. We investigate whether simple graph-dependent Instantaneous Quantum Polynomial-time (IQP) and Quantum Approximate Optimisation Algorithm (QAOA) circuits can distinguish non-isomorphic graphs. We first show that suitable IQP and QAOA constructions allow the underlying graph to be reconstructed using a polynomial number of samples. Building on previous work using the QAOA energy, we show that Hamming weight distributions can distinguish graph pairs at circuit depths for which the numerically computed energy gap is zero. We then apply graph invariants to the matrix of ZZ correlators on the output distribution. For the specific case of the graph spectrum, at QAOA depth p=2p=2p=2, this invariant numerically distinguishes all 10,48610{,}48610,486 cospectral regular graph pairs tested on up to 18 vertices. For IQP circuits, we introduce a finer invariant based on probability multiplicities within Hamming weight sectors, which distinguishes every non-isomorphic graph pair tested numerically. Finally, we derive exact graph-theoretic characterisations of the IQP output distribution at θ=π/2θ=π/2θ=π/2 and θ=π/4θ=π/4θ=π/4. These results show that simple graph-dependent quantum circuits encode substantial structural information, while leaving open whether this information can be extracted efficiently for general graph isomorphism.