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) on a graph with a set of unknown marked vertices, one can define a related absorbing walk where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian from the interpolated Markov chain 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.
- Received 19 April 2010
DOI:https://doi.org/10.1103/PhysRevA.82.022333
©2010 American Physical Society
Synopsis
Treasure hunt
Published 31 August 2010
A new approach proves that a quantum random walk is faster than the classical version at finding marked points on a graph.
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