Computational Complexity of Uniform Quantum Circuit Families and Quantum Turing Machines

TL;DR

This paper investigates the computational equivalence of quantum Turing machines and quantum circuit families, establishing their similarities for Monte Carlo algorithms and exploring open questions for Las Vegas algorithms.

Contribution

It formalizes the notions of codes and uniformity of QCFs, compares complexity classes of QTMs and QCFs, and proves the equivalence of various quantum Turing machine models.

Findings

Complexity classes for Monte Carlo algorithms are equivalent for QTMs and QCFs.

The equivalence of models for Las Vegas algorithms remains an open question.

Various types of QTMs are shown to be computationally equivalent.

Abstract

Deutsch proposed two sorts of models of quantum computers, quantum Turing machines (QTMs) and quantum circuit families (QCFs). In this paper we explore the computational powers of these models and re-examine the claim of the computational equivalence of these models often made in the literature without detailed investigations. For this purpose, we formulate the notion of the codes of QCFs and the uniformity of QCFs by the computability of the codes. Various complexity classes are introduced for QTMs and QCFs according to constraints on the error probability of algorithms or transition amplitudes. Their interrelations are examined in detail. For Monte Carlo algorithms, it is proved that the complexity classes based on uniform QCFs are identical with the corresponding classes based on QTMs. However, for Las Vegas algorithms, it is still open whether the two models are equivalent. We indicate the possibility that they are not equivalent. In addition, we give a complete proof of the existence of a universal QTM simulating multi-tape QTMs efficiently. We also examine the simulation of various types of QTMs such as multi-tape QTMs, single tape QTMs, stationary, normal form QTMs (SNQTMs), and QTMs with the binary tapes. As a result, we show that these QTMs are computationally equivalent one another as computing models implementing not only Monte Carlo algorithms but exact (or error-free) ones.