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arXiv:2610.19962

Testing graph isomorphism using simple quantum circuits

Samuel Beasley, Ashley Montanaro

quant-phSubmitted Oct 14, 2026• Announced Oct 15, 2026• 0 citations
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Abstract

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 $p=2$, this invariant numerically distinguishes all $10{,}486$ 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 $θ=π/2$ and $θ=π/4$. 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.

Comments
35 pages, 11 figures

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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.
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Graph isomorphism is an essential problem in graph theory and widely used in a variety of applications. With graph size increasing, graph isomorphism become difficult for classical computation since there is no polynomial algorithm for general graphs yet. Considering the advantages of quantum computation over classical computation in various tasks, quantum graph isomorphism algorithms have been studied to explore the potentials of quantum computation in graph isomorphism. In this work, we propose a graph isomorphism algorithm based on continuous-time quantum walks, where we construct the graph certificates from the output probabilities of continuous-time quantum walks on the tested graphs with added self-loops to graph vertices, and distinguish graphs by comparing the graph certificates. Numerous tests on diverse graph classes demonstrate the effectiveness of the proposed GI algorithm, which is not limited to a specific topological structure graph. Moreover, we test the proposed algorithm on classes of strongly regular graphs, and the results show that the distinguishing ability of the algorithm can be improved with the increasement of the added self-loops to distinct vertices.
Quantum generative models offer an inherently efficient sampling procedure, yet their training on quantum hardware is often a bottleneck. Parametrized Instantaneous Quantum Polynomial-time (IQP) circuits present a promising approach combining classical training with single-shot measurements for sample generation. However, its practical viability on Noisy Intermediate-Scale Quantum (NISQ) hardware remains an open question. In this work, we provide the first large-scale experimental validation of this method for learning graph distributions with both local (edge probabilities) and global (‘2-colorability’) constraints. While our graph encoding requires a number of qubits that scales quadratically with node count, our experiments, ranging from 28 to 153 qubits, demonstrate the approach's capabilities and limitations. For instance, a 153-qubit model successfully learned sparse ErdősRényi graphs with an average edge probability error below two percentage points. In contrast, learning the global property of bipartiteness proved more challenging and sensitive to noise, achieving 25.78 % accuracy for 18-node sparse instances. These results validate parametrized IQP circuits as a practical tool for generative modeling on near-term devices, while clearly delineating the critical challenges that must be addressed for future progress.
We propose a novel variational method for solving the sub-graph isomorphism problem on a gate-based quantum computer. The method relies (1) on a new representation of the adjacency matrices of the underlying graphs, which requires a number of qubits that scales logarithmically with the number of vertices of the graphs; and (2) on a new ansatz that can efficiently probe the permutation space. Simulations are then presented to showcase the approach on graphs up to 16 vertices, whereas, given the logarithmic scaling, the approach could be applied to realistic sub-graph isomorphism problem instances in the medium term.
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