Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Quantum Spectral Scattering for Graph Analysis

Vladislav Tishkov

Published Feb 4, 2026• 0 citations
Publisher

Abstract

This paper introduces a novel quantum algorithmic framework that achieves exponential speedup for fundamental graph problems, including Graph Isomorphism (GI), Maximum Clique, and Graph Coloring. The approach is based on spectral scattering operators \(S_G(\epsilon)\) derived from the spectral memory field \(\Phi_{\mathrm{QG}}\) and spectral curves \(X_G\). It is proven that the set of traces \(\{\mathrm{Tr}[S_G^k(\epsilon)]\}_{k=1}^n\) forms a complete graph invariant, enabling polynomial-time solutions to problems previously thought to require exponential or quasi-polynomial classical time. Key contributions include: 1. A quantum algorithm for GI with \(O(\mathrm{poly}(n))\) time and \(O(\log n)\) qubit requirements. 2. Extensions to NP-hard problems (Maximum Clique, Graph Coloring) with demonstrated quantum advantage. 3. Experimental validation on quantum simulators for graphs up to 64 vertices, achieving 100% accuracy. 4. Connections to quantum gravity via a holographic principle, analogizing graph encoding to AdS/CFT correspondence. The work provides full complexity analyses, error bounds, and open-source implementations, establishing a concrete quantum advantage while bridging quantum computation, spectral geometry, and quantum gravity.

DOI
10.5281/zenodo.18480996

Related papers

Quantum Algorithmic Complexity in Graph Theory -Shortest Path Problems and Quantum Acceleration Mechanisms

Jeffrey Quinn Cordova

We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.

•Oct 21, 2025•0 citations

A Quantum Framework for K Coloring of Graphs

Lord Sen, Shyamapada Mukherjee

Graph coloring is a well-known NP-complete problem with applications in scheduling, register allocation, frequency assignment, and network optimization. Quantum computing offers the potential for polynomial or even super-polynomial speedups in certain problem instances, yet practical quantum graph coloring methods must carefully balance qubit resources, circuit depth, and constraint enforcement. We proposed a solver agnostic quantum framework for $K$-coloring. Along with a novel encoding and efficient constraint implementation for its exact coloring exploiting Grover search and almost optimal coloring by the Quantum Approximate Optimization Algorithm \emph{(QAOA)}, and \emph{Quantum Annealing}. Unlike using $O(NK)$ qubits as in \emph{SOTA}, we reduced the qubit requirements to $O(N \log_2 K)$, along with optimized comparator circuits enforcing adjacency constraints. Further symmetry-based graph reduction is incorporated as an optional preprocessing step to further reduce the instance size before quantum execution.

quant-pharXiv:2609.32348•International Conference on Computational Intelligence and Communication Networks (CICN)•Sep 29, 2026•0 citations

Quantum Algorithms for Approximate Graph Isomorphism Testing

Prateek P. Kulkarni

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.

quant-pharXiv:2603.02656•Mar 4, 2026•0 citations

Quantum algorithms and the maximum clique problem in graphs

Matheus Augusto Condé Dias

Determining the cardinality of the maximum clique on a simple graph is a NP-hard problem with several applications of interest. The best know classical algorithm for this problem has 𝒪(3𝑛/3) time complexity. Classical computing uses phenomena from classical physics to model, store and process information. Consequently, the entire model of classical computing is bounded superiorly by the limits of classical physics. As an alternative, Quantum Computing has been developed to take advantage of quantum phenomena and provide the constructions of circuits capable of performing certain computations more efficiently. In this paper, two quantum solutions to the maximum clique problem will be proposed and analyzed, based on the Grover Quantum Search algorithms and the Quantum Minimal Search Algorithm, comparing their respective time complexities with the best known classical solution.

•Sep 14, 2022•0 citations

A Quantum-Inspired Algorithm for Graph Isomorphism

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

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.

quant-pharXiv:2512.24423•Jan 1, 2026•0 citations

Exponential-time quantum algorithms for graph coloring problems

Shimizu, Kazuya, Ryuhei Mori

The fastest known classical algorithm deciding the $k$-colorability of $n$-vertex graph requires running time $Ω(2^n)$ for $k\ge 5$. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time $O(1.9140^n)$ using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph $20$-coloring problem with running time $O(1.9575^n)$. In the polynomial-space quantum algorithm, we essentially show $(4-ε)^n$-time classical algorithms that can be improved quadratically by Grover's search.

arXiv:1907.00529•Jul 1, 2019•0 citations
This paper introduces a novel quantum algorithmic framework that achieves exponential speedup for fundamental graph problems, including Graph Isomorphism (GI), Maximum Clique, and Graph Coloring. The approach is based on spectral scattering operators (S_G(\epsilon)) derived from the spectral memory field (\Phi_{\mathrm{QG}}) and spectral curves (X_G). It is proven that the set of traces ({\mathrm{Tr}[S_G^k(\epsilon)]}_{k=1}^n) forms a complete graph invariant, enabling polynomial-time solutions to problems previously thought to require exponential or quasi-polynomial classical time. Key contributions include: 1. A quantum algorithm for GI with (O(\mathrm{poly}(n))) time and (O(\log n)) qubit requirements. 2. Extensions to NP-hard problems (Maximum Clique, Graph Coloring) with demonstrated quantum advantage. 3. Experimental validation on quantum simulators for graphs up to 64 vertices, achieving 100% accuracy. 4. Connections to quantum gravity via a holographic principle, analogizing graph encoding to AdS/CFT correspondence. The work provides full complexity analyses, error bounds, and open-source implementations, establishing a concrete quantum advantage while bridging quantum computation, spectral geometry, and quantum gravity.
We provide a rigorous theoretical analysis of quantum algorithms for shortest path problems in weighted graphs, establishing complexity-theoretic foundations for quantum acceleration in this fundamental computational domain. Building upon recent quantum adaptations of classical graph algorithms, we develop a unified framework that characterizes when and why quantum approaches offer computational advantages for single-source and all-pairs shortest path computations. Our analysis encompasses both positive-weight and negative-weight scenarios, deriving conditional speedup theorems that depend on graph structure, weight distribution, and oracle access models. We prove that quantum amplitude amplification and quantum walks can provide polynomial to exponential speedups under specific structural assumptions, while establishing hardness results that identify problem instances where quantum advantage is provably absent. The theoretical framework introduces novel complexity measures that capture graph-theoretic properties relevant to quantum algorithm performance, including quantum diameter, entanglement capacity, and interference efficiency. We extend our analysis to dynamic graph scenarios where edge weights evolve over time, establishing bounds on quantum advantage preservation under graph modifications. This work contributes fundamental insights into quantum algorithmic graph theory, clarifying the boundaries of quantum computational advantage and providing theoretical guidance for algorithm design in network optimization, routing, and graph analytics applications.
Graph coloring is a well-known NP-complete problem with applications in scheduling, register allocation, frequency assignment, and network optimization. Quantum computing offers the potential for polynomial or even super-polynomial speedups in certain problem instances, yet practical quantum graph coloring methods must carefully balance qubit resources, circuit depth, and constraint enforcement. We proposed a solver agnostic quantum framework for KKK-coloring. Along with a novel encoding and efficient constraint implementation for its exact coloring exploiting Grover search and almost optimal coloring by the Quantum Approximate Optimization Algorithm \emph{(QAOA)}, and \emph{Quantum Annealing}. Unlike using O(NK)O(NK)O(NK) qubits as in \emph{SOTA}, we reduced the qubit requirements to O(Nlog⁡2K)O(N \log_2 K)O(Nlog2​K), along with optimized comparator circuits enforcing adjacency constraints. Further symmetry-based graph reduction is incorporated as an optional preprocessing step to further reduce the instance size before quantum execution.
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 nnn vertices drawn from the Erdős--Rényi distribution G(n,1/2)\mathcal{G} (n,1/2)G(n,1/2) are considered approximately isomorphic if they can be made isomorphic by at most kkk edge edits. We present a quantum algorithm based on MNRS quantum walk search over the product graph Γ(G,H)Γ(G,H)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(n3/2log⁡n/ε)\mathcal{O}(n^{3/2} \log n/\varepsilon)O(n3/2logn/ε), where ε\varepsilonε parameterizes the approximation threshold. We complement this with an Ω(n2)Ω(n^2)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.
Determining the cardinality of the maximum clique on a simple graph is a NP-hard problem with several applications of interest. The best know classical algorithm for this problem has 𝒪(3𝑛/3) time complexity. Classical computing uses phenomena from classical physics to model, store and process information. Consequently, the entire model of classical computing is bounded superiorly by the limits of classical physics. As an alternative, Quantum Computing has been developed to take advantage of quantum phenomena and provide the constructions of circuits capable of performing certain computations more efficiently. In this paper, two quantum solutions to the maximum clique problem will be proposed and analyzed, based on the Grover Quantum Search algorithms and the Quantum Minimal Search Algorithm, comparing their respective time complexities with the best known classical solution.
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.
The fastest known classical algorithm deciding the kkk-colorability of nnn-vertex graph requires running time Ω(2n)Ω(2^n)Ω(2n) for k≥5k\ge 5k≥5. In this work, we present an exponential-space quantum algorithm computing the chromatic number with running time O(1.9140n)O(1.9140^n)O(1.9140n) using quantum random access memory (QRAM). Our approach is based on Ambainis et al's quantum dynamic programming with applications of Grover's search to branching algorithms. We also present a polynomial-space quantum algorithm not using QRAM for the graph 202020-coloring problem with running time O(1.9575n)O(1.9575^n)O(1.9575n). In the polynomial-space quantum algorithm, we essentially show (4−ε)n(4-ε)^n(4−ε)n-time classical algorithms that can be improved quadratically by Grover's search.