Quantum walks: a comprehensive review

TL;DR

This comprehensive review covers the theoretical foundations, models, algorithms, and experimental progress of quantum walks, highlighting their role as a universal model for quantum computation and their potential in quantum algorithm development.

Contribution

It provides an extensive synthesis of recent advances in quantum walks, including their mathematical models, role of randomness, entanglement, and their universality in quantum computation.

Findings

Quantum walks are a universal model of quantum computation.

Theoretical frameworks for discrete and continuous quantum walks are well-developed.

Experimental proposals and realizations of quantum walks are summarized.

Abstract

Quantum walks, the quantum mechanical counterpart of classical random walks, is an advanced tool for building quantum algorithms that has been recently shown to constitute a universal model of quantum computation. Quantum walks is now a solid field of research of quantum computation full of exciting open problems for physicists, computer scientists, mathematicians and engineers. In this paper we review theoretical advances on the foundations of both discrete- and continuous-time quantum walks, together with the role that randomness plays in quantum walks, the connections between the mathematical models of coined discrete quantum walks and continuous quantum walks, the quantumness of quantum walks, a summary of papers published on discrete quantum walks and entanglement as well as a succinct review of experimental proposals and realizations of discrete-time quantum walks. Furthermore, we have reviewed several algorithms based on both discrete- and continuous-time quantum walks as well as a most important result: the computational universality of both continuous- and discrete- time quantum walks.