Universal computation by quantum walk

TL;DR

This paper demonstrates that universal quantum computation can be achieved through quantum walks on sparse graphs, encoding quantum gates via scattering processes, thus simplifying the Hamiltonian requirements.

Contribution

It shows that even with a sparse adjacency matrix, quantum walks can implement universal quantum computation, expanding the understanding of quantum computational primitives.

Findings

Quantum walks on sparse graphs are universal for quantum computation.

Quantum gates can be implemented through scattering processes in the graph.

Universal computation is possible with Hamiltonians having entries only 0 or 1.

Abstract

In some of the earliest work on quantum mechanical computers, Feynman showed how to implement universal quantum computation by the dynamics of a time-independent Hamiltonian. I show that this remains possible even if the Hamiltonian is restricted to be a sparse matrix with all entries equal to 0 or 1, i.e., the adjacency matrix of a low-degree graph. Thus quantum walk can be regarded as a universal computational primitive, with any desired quantum computation encoded entirely in some underlying graph. The main idea of the construction is to implement quantum gates by scattering processes.