Quantum Walk on the Line

TL;DR

This paper analyzes the behavior of quantum walks on the line, demonstrating faster mixing times compared to classical walks, with implications for quantum algorithms and graph analysis.

Contribution

It provides a detailed analysis of the unbiased Hadamard quantum walk on the line, showing its rapid mixing properties and potential for application to general graphs.

Findings

Quantum walk distribution becomes almost uniform over a line segment after t steps.

Quantum walks on a circle mix in linear time, faster than classical counterparts.

Techniques can be extended to analyze quantum walks on more complex graphs.

Abstract

Motivated by the immense success of random walk and Markov chain methods in the design of classical algorithms, we consider_quantum_ walks on graphs. We analyse in detail the behaviour of unbiased quantum walk on the line, with the example of a typical walk, the ``Hadamard walk''. We show that after t time steps, the probability distribution on the line induced by the Hadamard walk is almost uniformly distributed over the interval [-t/sqrt(2),t/sqrt(2)]. This implies that the same walk defined on the circle mixes in_linear_ time. This is in direct contrast with the quadratic mixing time for the corresponding classical walk. We conclude by indicating how our techniques may be applied to more general graphs.