Limitation of Quantum Walk Approach to the Maximum Matching Problem

• URL: http://arxiv.org/abs/2510.26246v1
• Date: Thu, 30 Oct 2025 08:29:44 GMT
• Title: Limitation of Quantum Walk Approach to the Maximum Matching Problem
• Authors: Alcides Gomes Andrade Júnior, Akira Matsubayashi,
• Abstract summary: The Maximum Matching problem has a quantum query complexity lower bound of $Omega(n3/2)$ for graphs on $n$ vertices represented by an adjacency matrix.<n>The current best quantum algorithm has the query complexity $O(n7/4)$, which is an improvement over the trivial bound $O(n2)$.<n>We show that the quantum walk technique fails to produce a fast algorithm improving the known (or even the trivial) upper bound on the query complexity.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The Maximum Matching problem has a quantum query complexity lower bound of $\Omega(n^{3/2})$ for graphs on $n$ vertices represented by an adjacency matrix. The current best quantum algorithm has the query complexity $O(n^{7/4})$, which is an improvement over the trivial bound $O(n^2)$. Constructing a quantum algorithm for this problem with a query complexity improving the upper bound $O(n^{7/4})$ is an open problem. The quantum walk technique is a general framework for constructing quantum algorithms by transforming a classical random walk search into a quantum search, and has been successfully applied to constructing an algorithm with a tight query complexity for another problem. In this work we show that the quantum walk technique fails to produce a fast algorithm improving the known (or even the trivial) upper bound on the query complexity. Specifically, if a quantum walk algorithm designed with the known technique solves the Maximum Matching problem using $O(n^{2-\epsilon})$ queries with any constant $\epsilon>0$, and if the underlying classical random walk is independent of an input graph, then the guaranteed time complexity is larger than any polynomial of $n$.

Related papers

• Pseudo-deterministic Quantum Algorithms [7.46931129146594]
  We show that for any total problem $R$, pseudo-deterministic quantum algorithms admit at most a quintic advantage over deterministic algorithms.<n>On the algorithmic side, we identify a class of quantum search problems that can be made pseudo-deterministic with small overhead.
  arXiv  Detail & Related papers  (2026-02-19T18:54:47Z)
• Qudit-based scalable quantum algorithm for solving the integer programming problem [0.0]
  programming (IP) is an NP-hard optimization problem that is widely used to represent a diverse set of real-world problems.<n>Most quantum algorithms for solving IP are highly resource inefficient because they encode integers into qubits.<n>In this work, a circuit-based scalable quantum algorithm is presented using multiple interacting qudits for which we show a quantum speed-up.
  arXiv  Detail & Related papers  (2025-08-19T15:06:49Z)
• Space-Efficient Quantum Error Reduction without log Factors [50.10645865330582]
  We present a new highly simplified construction of a purifier, that can be understood as a weighted walk on a line similar to a random walk interpretation of majority voting.<n>Our purifier has exponentially better space complexity than the previous one, and quadratically better dependence on the soundness-completeness gap of the algorithm being purified.
  arXiv  Detail & Related papers  (2025-02-13T12:04:39Z)
• Sum-of-Squares inspired Quantum Metaheuristic for Polynomial Optimization with the Hadamard Test and Approximate Amplitude Constraints [76.53316706600717]
  Recently proposed quantum algorithm arXiv:2206.14999 is based on semidefinite programming (SDP) We generalize the SDP-inspired quantum algorithm to sum-of-squares. Our results show that our algorithm is suitable for large problems and approximate the best known classicals.
  arXiv  Detail & Related papers  (2024-08-14T19:04:13Z)
• Quantum algorithms for Hopcroft's problem [45.45456673484445]
  We study quantum algorithms for Hopcroft's problem which is a fundamental problem in computational geometry. The classical complexity of this problem is well-studied, with the best known algorithm running in $O(n4/3)$ time. Our results are two different quantum algorithms with time complexity $widetilde O(n5/6)$.
  arXiv  Detail & Related papers  (2024-05-02T10:29:06Z)
• Taming Quantum Time Complexity [45.867051459785976]
  We show how to achieve both exactness and thriftiness in the setting of time complexity. We employ a novel approach to the design of quantum algorithms based on what we call transducers.
  arXiv  Detail & Related papers  (2023-11-27T14:45:19Z)
• On the exact quantum query complexity of $\text{MOD}_m^n$ and $\text{EXACT}_{k,l}^n$ [4.956977275061968]
  We present an exact quantum algorithm for computing $textMOD_mn$. We show exact quantum query complexity of a broad class of symmetric functions that map $0,1n$ to a finite set $X$ is less than $n$.
  arXiv  Detail & Related papers  (2023-03-20T08:17:32Z)
• Quantum Worst-Case to Average-Case Reductions for All Linear Problems [66.65497337069792]
  We study the problem of designing worst-case to average-case reductions for quantum algorithms. We provide an explicit and efficient transformation of quantum algorithms that are only correct on a small fraction of their inputs into ones that are correct on all inputs.
  arXiv  Detail & Related papers  (2022-12-06T22:01:49Z)
• Variational learning algorithms for quantum query complexity [3.980076328494117]
  We develop variational learning algorithms to study quantum query complexity. We apply our method to analyze various cases of quantum query complexity. Our method can be readily implemented on the near-term Noisy Intermediate-Scale Quantum (NISQ) devices.
  arXiv  Detail & Related papers  (2022-05-16T05:16:15Z)
• On Applying the Lackadaisical Quantum Walk Algorithm to Search for Multiple Solutions on Grids [63.75363908696257]
  The lackadaisical quantum walk is an algorithm developed to search graph structures whose vertices have a self-loop of weight $l$. This paper addresses several issues related to applying the lackadaisical quantum walk to search for multiple solutions on grids successfully.
  arXiv  Detail & Related papers  (2021-06-11T09:43:09Z)
• Time and Query Optimal Quantum Algorithms Based on Decision Trees [2.492300648514128]
  We show that a quantum algorithm can be implemented in time $tilde O(sqrtGT)$. Our algorithm is based on non-binary span programs and their efficient implementation.
  arXiv  Detail & Related papers  (2021-05-18T06:51:11Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.