On the adiabatic condition and the quantum hitting time of Markov chains

Details On the adiabatic condition and the quantum hitting time of Markov chains On the adiabatic condition and the quantum hitting time of Markov chains

2010-04-15

arxiv.org/abs/1004.2721v1

Physical Review A, 82(2):022333, 2010

Physics

Quantum Physics

22 pages

Scientific paper

10.1103/PhysRevA.82.022333

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 $P'$ where outgoing transitions from marked vertices are replaced by self-loops. We build a Hamiltonian $H(s)$ from the interpolated Markov chain $P(s)=(1-s)P+sP'$ 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 speed-up for state-transitive reversible Markov chains with a unique marked vertex.

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