Exceptional quantum walk search on the cycle

  • Thomas G. WongEmail author
  • Raqueline A. M. Santos


Quantum walks are standard tools for searching graphs for marked vertices, and they often yield quadratic speedups over a classical random walk’s hitting time. In some exceptional cases, however, the system only evolves by sign flips, staying in a uniform probability distribution for all time. We prove that the one-dimensional periodic lattice or cycle with any arrangement of marked vertices is such an exceptional configuration. Using this discovery, we construct a search problem where the quantum walk’s random sampling yields an arbitrary speedup in query complexity over the classical random walk’s hitting time. In this context, however, the mixing time to prepare the initial uniform state is a more suitable comparison than the hitting time, and then, the speedup is roughly quadratic.


Quantum walk Quantum search Spatial search Exceptional configuration Random walk Markov chain Hitting time Mixing time 



T.W. thanks the quantum computing group at the University of Texas at Austin for useful discussions. T.W. was supported by the U.S. Department of Defense Vannevar Bush Faculty Fellowship of Scott Aaronson. R.S. was supported by the RAQUEL (Grant Agreement No. 323970) project and the ERC Advanced Grant MQC.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of Texas at AustinAustinUSA
  2. 2.Center for Quantum Computer ScienceUniversity of LatviaRigaLatvia

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