Efficient quantum circuit implementation of quantum walks

TL;DR

This paper explores the design of efficient quantum circuits for implementing quantum walks on symmetric graphs, aiming to achieve exponential speedups in quantum search algorithms without relying on oracles.

Contribution

It introduces methods for constructing efficient quantum circuits for quantum walks on symmetric graphs, expanding the potential for oracle-free quantum speedups.

Findings

Efficient quantum circuits can be constructed for quantum walks on certain symmetric graphs.

Potential for exponential quantum speedup in search algorithms without oracles.

Discussion of applications to quantum search problems.

Abstract

Quantum walks, being the quantum analogue of classical random walks, are expected to provide a fruitful source of quantum algorithms. A few such algorithms have already been developed, including the `glued trees' algorithm, which provides an exponential speedup over classical methods, relative to a particular quantum oracle. Here, we discuss the possibility of a quantum walk algorithm yielding such an exponential speedup over possible classical algorithms, without the use of an oracle. We provide examples of some highly symmetric graphs on which efficient quantum circuits implementing quantum walks can be constructed, and discuss potential applications to quantum search for marked vertices along these graphs.