We do not claim to solve the general graph automorphism problem. Whileit is a known strategy to attempt to solve the Graph isomorphism and autom-rphism problems on a quantum computer by treating them as a HSP oversome symmetric group this paper takes a different approach by using the graphformlism presented in [1] so we present a intrisically quantum algorithm for ex-ploring the local-Clifford (LC) equivalence neighborhood of a graph using thegraph-state data structure introduced by [1]. By leveraging efficient local com-plementation, edge complementation, and ancilla-driven set-complementationoperations, we construct a quantum superposition over the LC-orbit of a graphand introduce an oracle that implements structural invariants such as automor-phisms and fixed-point subgraphs. We show that amplitude amplification yieldsa quadratic speedup for detecting LC-invariant features compared to classicalrandomized search. Our method provides a provably correct quantum proce-dure for structural graph analysis within the graph-state formalism. The localCliford orbit is a massive, exponentially large space that cannot be searched ingeneral classically with polynomial time complexity
We do not claim to solve the general graph automorphism problem. Whileit is a known strategy to attempt to solve the Graph isomorphism and autom-rphism problems on a quantum computer by treating them as a HSP oversome symmetric group this paper takes a different approach by using the graphformlism presented in [1] so we present a intrisically quantum algorithm for ex-ploring the local-Clifford (LC) equivalence neighborhood of a graph using thegraph-state data structure introduced by [1]. By leveraging efficient local com-plementation, edge complementation, and ancilla-driven set-complementationoperations, we construct a quantum superposition over the LC-orbit of a graphand introduce an oracle that implements structural invariants such as automor-phisms and fixed-point subgraphs. We show that amplitude amplification yieldsa quadratic speedup for detecting LC-invariant features compared to classicalrandomized search. Our method provides a provably correct quantum proce-dure for structural graph analysis within the graph-state formalism. The localCliford orbit is a massive, exponentially large space that cannot be searched ingeneral classically with polynomial time complexity
In Van den Nest et al. [Phys. Rev. A 69, 022316 (2004)] we presented a description of the action of local Clifford operations on graph states in terms of a graph transformation rule, known in graph theory as local complementation. It was shown that two graph states are equivalent under the local Clifford group if and only if there exists a sequence of local complementations which relates their associated graphs. In this Brief Report we report the existence of a polynomial time algorithm, published in A. Bouchet [Combinatorica 11, 315 (1991)], which decides whether two given graphs are related by a sequence of local complementations. Hence an efficient algorithm to detect local Clifford equivalence of graph states is obtained.
Graph partitioning is a fundamental problem in computer science with applications in diverse fields like network design, compiler optimization, and data analysis. Traditional algorithms for graph partitioning often suffer from computational complexity, particularly when dealing with large graphs. This paper explores the potential of utilizing quantum computing to accelerate graph partitioning. Specifically, we investigate the application of Grover's algorithm, a quantum search algorithm, to the problem. Our core claim is that by leveraging the superposition and interference capabilities inherent in quantum mechanics, a quantum solution can achieve a provable speedup compared to classical approaches. We present a theoretical framework outlining the algorithm's operation and derive the expected speedup. The key mathematical formulation involves representing the graph partitioning problem as an unstructured search problem amenable to Grover's algorithm. We define the problem as minimizing the cut size, represented by the objective function: *f(S) = Σi∈S Σj∈Sc d(i, j)* where *S* is a subset of vertices, *Sc* is the complement of *S* in the graph, and *d(i, j)* is the shortest path distance between vertices *i* and *j*. The algorithm then constructs a quantum oracle that maps the problem state to a superposition of all possible partitions. Grover's algorithm is then applied to amplify the amplitude of the solution state, resulting in a high probability of finding an optimal or near-optimal partition. The speedup is directly proportional to the square root of the number of iterations performed by Grover's algorithm, which is influenced by the problem size and the quality of the oracle. We analyze the complexity of the quantum algorithm and compare it to the complexity of the best-known classical algorithms, demonstrating a potential for significant speedup. Further research will focus on optimizing the oracle construction and exploring other quantum algorithms suitable for graph partitioning.
Graph states and their entanglement properties are pivotal for the development of quantum computing and technologies. For qubits, local complementation, a graphical rule that connects all the equivalent states under Local Clifford (LC) operations, was used for the complete characterization of all the LC equivalence classes up until 12 particles, assisting applications in quantum error correction and state preparation protocols optimization. This concept has been extended for qudits. In this work, we provide a complete characterization of the entanglement classes up until 7 qutrits, mapping each class into an orbit. The graph-theoretic properties of the orbits are studied, illuminating the rich structure they have. Clear connections between the connectivity of the orbits and the entanglement properties are observed. The correlations between the graph-theoretic properties and the Schmidt measure provide useful insights regarding qudit state preparation and fault-tolerance. This work is accompanied by a repository consisting of the code to extract the orbits and perform the presented statistical analysis. The strong interplay between quantum theory and graph theory is well-known and extensively studied for qubits. Our work provides practical tools and results to assist this endeavor for the qudit case.
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.
We present an algorithm for verifying the local unitary (LU) equivalence of graph and stabilizer states. Our approach reduces the problem to solving a system of linear equations in modular arithmetic. Furthermore, we demonstrate that any LU transformation between two graph states takes a specific form, naturally generalizing the class of local Clifford (LC) transformations. Lastly, using existing libraries, we verify that for up to n=11, the number of LU and LC orbits of stabilizer states is identical.
Graph states form a large family of quantum states that are in one-to-one correspondence with mathematical graphs. Graph states are used in many applications, such as measurement-based quantum computation, as multipartite entangled resources. It is thus crucial to understand when two such states have the same entanglement, i.e. when they can be transformed into each other using only local operations. In this case, we say that the graph states are LU-equivalent (local unitary). If the local operations are restricted to the so-called Clifford group, we say that the graph states are LC-equivalent (local Clifford). Interestingly, a simple graph rule called local complementation fully captures LC-equivalence, in the sense that two graph states are LC-equivalent if and only if the underlying graphs are related by a sequence of local complementations. While it was once conjectured that two LU-equivalent graph states are always LC-equivalent, counterexamples do exist and local complementation fails to fully capture the entanglement of graph states. We introduce in this thesis a generalization of local complementation that does fully capture LU-equivalence. Using this characterization, we prove the existence of an infinite strict hierarchy of local equivalences between LC- and LU-equivalence. This also leads to the design of a quasi-polynomial algorithm for deciding whether two graph states are LU-equivalent, and to a proof that two LU-equivalent graph states are LC-equivalent if they are defined on at most 19 qubits. Furthermore, we study graph states that are universal in the sense that any smaller graph state, defined on any small enough set of qubits, can be induced using only local operations. We provide bounds and an optimal, probabilistic construction.