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Title: Adiabatic condition and the quantum hitting time of Markov chains

Abstract

We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) P on a graph with a set of unknown marked vertices, one can define a related absorbing walk P{sup '} where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s) from the interpolated Markov chain P(s)=(1-s)P+sP{sup '} and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that, for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speedup for state-transitive reversible Markov chains with a unique marked vertex.

Authors:
 [1];  [1];  [1]
  1. NEC Laboratories America, Inc., Princeton, New Jersey 08540 (United States)
Publication Date:
OSTI Identifier:
21448480
Resource Type:
Journal Article
Journal Name:
Physical Review. A
Additional Journal Information:
Journal Volume: 82; Journal Issue: 2; Other Information: DOI: 10.1103/PhysRevA.82.022333; (c) 2010 The American Physical Society; Journal ID: ISSN 1050-2947
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGORITHMS; GRAPH THEORY; HAMILTONIANS; MARKOV PROCESS; QUANTUM STATES; RANDOMNESS; MATHEMATICAL LOGIC; MATHEMATICAL OPERATORS; MATHEMATICS; QUANTUM OPERATORS; STOCHASTIC PROCESSES

Citation Formats

Krovi, Hari, University of Connecticut, Storrs, Connecticut 06269, Ozols, Maris, University of Waterloo and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, and Roland, Jeremie. Adiabatic condition and the quantum hitting time of Markov chains. United States: N. p., 2010. Web. doi:10.1103/PHYSREVA.82.022333.
Krovi, Hari, University of Connecticut, Storrs, Connecticut 06269, Ozols, Maris, University of Waterloo and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, & Roland, Jeremie. Adiabatic condition and the quantum hitting time of Markov chains. United States. https://doi.org/10.1103/PHYSREVA.82.022333
Krovi, Hari, University of Connecticut, Storrs, Connecticut 06269, Ozols, Maris, University of Waterloo and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1, and Roland, Jeremie. 2010. "Adiabatic condition and the quantum hitting time of Markov chains". United States. https://doi.org/10.1103/PHYSREVA.82.022333.
@article{osti_21448480,
title = {Adiabatic condition and the quantum hitting time of Markov chains},
author = {Krovi, Hari and University of Connecticut, Storrs, Connecticut 06269 and Ozols, Maris and University of Waterloo and Institute for Quantum Computing, Waterloo, Ontario, N2L 3G1 and Roland, Jeremie},
abstractNote = {We present an adiabatic quantum algorithm for the abstract problem of searching marked vertices in a graph, or spatial search. Given a random walk (or Markov chain) P on a graph with a set of unknown marked vertices, one can define a related absorbing walk P{sup '} where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian H(s) from the interpolated Markov chain P(s)=(1-s)P+sP{sup '} and use it in an adiabatic quantum algorithm to drive an initial superposition over all vertices to a superposition over marked vertices. The adiabatic condition implies that, for any reversible Markov chain and any set of marked vertices, the running time of the adiabatic algorithm is given by the square root of the classical hitting time. This algorithm therefore demonstrates a novel connection between the adiabatic condition and the classical notion of hitting time of a random walk. It also significantly extends the scope of previous quantum algorithms for this problem, which could only obtain a full quadratic speedup for state-transitive reversible Markov chains with a unique marked vertex.},
doi = {10.1103/PHYSREVA.82.022333},
url = {https://www.osti.gov/biblio/21448480}, journal = {Physical Review. A},
issn = {1050-2947},
number = 2,
volume = 82,
place = {United States},
year = {Sun Aug 15 00:00:00 EDT 2010},
month = {Sun Aug 15 00:00:00 EDT 2010}
}