TY - JOUR

T1 - UNSTRUCTURED SEARCH BY RANDOM AND QUANTUM WALK

AU - Wong, Thomas G.

N1 - Funding Information:
Thanks to Zak Webb for pointing out the spectral norm in the continuous-time random walk. The early stages of this work were carried out while the author was a postdoctoral scholar at the University of Texas at Austin under Scott Aaronson, so this work was partially supported by the U.S. Department of Defense Vannevar Bush Faculty Fellowship of Scott Aaronson.
Publisher Copyright:
© Rinton Press.

PY - 2022/1

Y1 - 2022/1

N2 - 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 π√N/2√ 2 timesteps to reach a success probability of 1/2, while the continuous-time quantum walk takes π√N/2 time to reach a success probability of 1.

AB - 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 π√N/2√ 2 timesteps to reach a success probability of 1/2, while the continuous-time quantum walk takes π√N/2 time to reach a success probability of 1.

UR - http://www.scopus.com/inward/record.url?scp=85124606660&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85124606660&partnerID=8YFLogxK

U2 - 10.26421/QIC22.1-2-4

DO - 10.26421/QIC22.1-2-4

M3 - Article

AN - SCOPUS:85124606660

VL - 22

SP - 53

EP - 85

JO - Quantum Information and Computation

JF - Quantum Information and Computation

SN - 1533-7146

IS - 1-2

ER -