The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and pattern recognition require a flexible notion of structural similarity. We study the quantum query complexity of approximate graph isomorphism testing, where two graphs on $n$ vertices drawn from the Erdős--Rényi distribution $\mathcal{G} (n,1/2)$ are considered approximately isomorphic if they can be made isomorphic by at most $k$ edge edits.
We present a quantum algorithm based on MNRS quantum walk search over the product graph $Γ(G,H)$ of the two input graphs. When the graphs are approximately isomorphic, the quantum walk search detects vertex pairs belonging to a dense near isomorphic matching set; candidate pairings are then reconstructed via local consistency propagation and verified via a Grover-accelerated consistency check. We prove that this approach achieves query complexity $\mathcal{O}(n^{3/2} \log n/\varepsilon)$, where $\varepsilon$ parameterizes the approximation threshold. We complement this with an $Ω(n^2)$ classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model.
We extend the framework to spectral similarity measures based on graph Laplacian eigenvalues, as well as weighted and attributed graphs. Small-scale simulation results on quantum simulators for graphs with up to twenty vertices demonstrate compatibility with near-term quantum devices.
The graph isomorphism problem asks whether two graphs are identical up to vertex relabeling. While the exact problem admits quasi-polynomial-time classical algorithms, many applications in molecular comparison, noisy network analysis, and pattern recognition require a flexible notion of structural similarity. We study the quantum query complexity of approximate graph isomorphism testing, where two graphs on n vertices drawn from the Erdős--Rényi distribution G(n,1/2) are considered approximately isomorphic if they can be made isomorphic by at most k edge edits.
We present a quantum algorithm based on MNRS quantum walk search over the product graph Γ(G,H) of the two input graphs. When the graphs are approximately isomorphic, the quantum walk search detects vertex pairs belonging to a dense near isomorphic matching set; candidate pairings are then reconstructed via local consistency propagation and verified via a Grover-accelerated consistency check. We prove that this approach achieves query complexity O(n3/2logn/ε), where ε parameterizes the approximation threshold. We complement this with an Ω(n2) classical lower bound for constant approximation, establishing a genuine polynomial quantum speedup in the query model.
We extend the framework to spectral similarity measures based on graph Laplacian eigenvalues, as well as weighted and attributed graphs. Small-scale simulation results on quantum simulators for graphs with up to twenty vertices demonstrate compatibility with near-term quantum devices.
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.
In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that Θ(N) queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where N denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform O(N1/3) queries, showing a polynomial speedup. They also proved that Ω(N1/4) queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight Ω(N1/3) quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.
The quantum query complexity of subgraph-containment problems, which ask whether a given subgraph H is present in an input graph G, has been the subject of considerable study. However, even for relatively simple subgraphs, such as paths and cycles, a complete understanding of their query complexities remains elusive. In this work, we consider several variants of path- and cycle-containment problems in the adjacency matrix model, where we search for paths or cycles of constant length k. We compare the settings where the graphs are directed or undirected, where the goal is to detect or find the existence of a path/cycle, and where the path/cycle we are looking for has length exactly k, or at most k. We also consider several promise versions of these problems, where we suppose that the input graph has a certain structure. We characterize the relative difficulty of these variants of the path/cycle-containment problems, by relating them to one another using randomized reductions, and grouping them into equivalence classes.
When we restrict our attention to path-containment problems, we get a dichotomy result. Some of the path-containment problems can be solved using a linear number of queries, and all the others are equivalent to one another (and additionally to several cycle-containment problems) under randomized reductions, up to constant overhead. For the latter equivalence class, we prove a novel quantum-walk-based algorithm that achieves query complexity O(n3/2−αk), where αk∈Θ(c−k) and c=3+17/2≈1.33, beating the previous best upper bound O(n3/2) on its query complexity. We also provide a conditional lower bound based on the graph-collision problem, which implies that this equivalence class does not admit linear-query quantum algorithms unless graph collision admits an O(n) query algorithm.
The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an N-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every K≥2 and N≥4K−1, finds and verifies a homogeneous K-set using O(2KKlogηK) edge queries with failure probability at most η. At the Ramsey scale N=2n, this yields a homogeneous set of order ⌊n/2⌋+1 using O(NlogNlog(logN/η)) queries, improving on the O(N) queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an Ω(N1/12) quantum lower bound by a reduction from collision finding.
The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an Ω(N1−1/2) randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution G(N,1/2). On that distribution a greedy quantum search uses only O(N1/4) queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.
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.
Abstract This paper introduces a novel alternated two-particle discrete-time quantum walk model on arbitrary graphs, overcoming the limitation that conventional shared coin schemes are confined to regular graphs. By employing a fixed-dimension coin operator and encoding the graph structural information into the shift operator, the new model becomes applicable to arbitrary graphs. Further, by adding interaction between the two particles, an extension is presented, which can enhance the model’s dynamic properties and facilitate more intricate quantum interference phenomena. Based on the new model, a kind of quantum graph isomorphism algorithm framework is proposed. It just need O(N 0.5*∣E∣) steps of quantum walk and O(N 2) dimensions of Hilbert space, which offers a significant reduction in complexity compared to other quantum walk based algorithms. The graph isomorphism testing is performed in the framework using non-interacting and interacting quantum walks respectively. Experimental results illustrate that the algorithm based on interacting particles achieves a 100% success rate in discriminating all test graphs. While the algorithm with non-interacting particles performs a 100% distinction success rate on general non-regular graphs, albeit being ineffective on strongly regular graphs. Due to the effectiveness of the algorithm on arbitrary graphs, the algorithm has broad application prospects in the identification and comparison of chemical molecular structures, the analysis of social networks and so on.