Quantum Walk Sampling by Growing Seed Sets

Author Simon Apers

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Simon Apers
  • Inria, Paris, France
  • CWI, Amsterdam, The Netherlands


This work benefited from discussions with Alain Sarlette, Stacey Jeffery, Anthony Leverrier, Ronald de Wolf, André Chailloux and Frédéric Magniez. Part of this work was supported by the CWI-Inria International Lab.

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Simon Apers. Quantum Walk Sampling by Growing Seed Sets. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 9:1-9:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)


This work describes a new algorithm for creating a superposition over the edge set of a graph, encoding a quantum sample of the random walk stationary distribution. The algorithm requires a number of quantum walk steps scaling as O~(m^(1/3) delta^(-1/3)), with m the number of edges and delta the random walk spectral gap. This improves on existing strategies by initially growing a classical seed set in the graph, from which a quantum walk is then run. The algorithm leads to a number of improvements: (i) it provides a new bound on the setup cost of quantum walk search algorithms, (ii) it yields a new algorithm for st-connectivity, and (iii) it allows to create a superposition over the isomorphisms of an n-node graph in time O~(2^(n/3)), surpassing the Omega(2^(n/2)) barrier set by index erasure.

Subject Classification

ACM Subject Classification
  • Theory of computation → Quantum computation theory
  • Theory of computation → Graph algorithms analysis
  • Quantum algorithms
  • Quantum walks
  • Connectivity
  • Graph theory


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