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The task of finding an entry in an unsorted list of N elements famously takes O(N) queries to an oracle for a classical computer and O( N) queries for a quantum computer using Grover’s algorithm. Reformulated as a spatial search problem, this corresponds to searching the complete graph, or all-to-all network, for a marked vertex by querying an oracle. In this tutorial, we derive how discrete-and continuous-time (classical) random walks and quantum walks solve this problem in a thorough and pedagogical manner, providing an accessible introduction to how random and quantum walks can be used to search spatial regions. Some of the results are already known, but many are new. For large N, the random walks converge to the same evolution, both taking N ln(1/ɛ) time to reach a success probability of 1 − ɛ. In contrast, the discrete-time quantum walk asymptotically takes πN/2 2 timesteps to reach a success probability of 1/2, while the continuous-time quantum walk takes πN/2 time to reach a success probability of 1.

Original languageEnglish (US)
Pages (from-to)53-85
Number of pages33
JournalQuantum Information and Computation
Issue number1-2
StatePublished - Jan 2022

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Computational Theory and Mathematics


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