Universality and programmability of quantum computers

TL;DR

This paper examines the concept of universality and programmability in quantum computers, analyzing how quantum logic circuits and Turing machines relate to universal computation and identifying open problems in the field.

Contribution

It provides a detailed analysis of the different notions of universality in quantum computing and scrutinizes the extent to which these concepts have been established.

Findings

Quantum logic circuits can simulate arbitrary unitary transformations.

The universality of quantum Turing machines remains an open problem.

Different notions of universality have distinct implications for programmability.

Abstract

Manin, Feynman, and Deutsch have viewed quantum computing as a kind of universal physical simulation procedure. Much of the writing about quantum logic circuits and quantum Turing machines has shown how these machines can simulate an arbitrary unitary transformation on a finite number of qubits. The problem of universality has been addressed most famously in a paper by Deutsch, and later by Bernstein and Vazirani as well as Kitaev and Solovay. The quantum logic circuit model, developed by Feynman and Deutsch, has been more prominent in the research literature than Deutsch's quantum Turing machines. Quantum Turing machines form a class closely related to deterministic and probabilistic Turing machines and one might hope to find a universal machine in this class. A universal machine is the basis of a notion of programmability. The extent to which universality has in fact been established by the pioneers in the field is examined and this key notion in theoretical computer science is scrutinised in quantum computing by distinguishing various connotations and concomitant results and problems.