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

TL;DR

This paper introduces an adiabatic quantum algorithm for spatial search on graphs, establishing a new link between the adiabatic condition and the classical hitting time of Markov chains, and extending quantum speed-up to broader cases.

Contribution

It presents a novel adiabatic quantum algorithm that relates running time to classical hitting time, broadening the scope beyond previous algorithms limited to specific Markov chains.

Findings

The quantum algorithm's runtime is proportional to the square root of the classical hitting time.

It extends quantum search algorithms to non-state-transitive Markov chains with multiple marked vertices.

A new connection between adiabatic conditions and classical random walk properties is demonstrated.

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 $P'$ 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.