Umesh V. Vazirani

Quantum Complexity Theory

In this paper we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing machine in Deutschs model of a quantum Turing machine (QTM) [Proc. Roy. Soc. London Ser. A, 400 (1985), pp. 97--117]. This construction is substantially more complicated than the corresponding construction for classical Turing machines (TMs); in fact, even simple primitives such as looping, branching, and composition are not straightforward in the context of quantum Turing machines. We establish how these familiar primitives can be implemented and introduce some new, purely quantum mechanical primitives, such as changing the computational basis and carrying out an arbitrary unitary transformation of polynomially bounded dimension. We also consider the precision to which the transition amplitudes of a quantum Turing machine need to be specified. We prove that

Strengths and Weaknesses of Quantum Computing

O(\log T)

Matching is as easy as matrix inversion

bits of precision suffice to support a

AdWords and generalized online matching

T

Quantum complexity theory

step computation. This justifies the claim that the quantum Turing machine model should be regarded as a discrete model of computation and not an analog one. We give the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis. We show the existence of a problem, relative to an oracle, that can be solved in polynomial time on a quantum Turing machine, but requires superpolynomial time on a bounded-error probabilistic Turing machine, and thus not in the class

Quantum walks on graphs

\BPP

Generating quasi-random sequences from semi-random sources

. The class

AdWords and generalized on-line matching

\BQP

Fully Device-Independent Quantum Key Distribution

of languages that are efficiently decidable (with small error-probability) on a quantum Turing machine satisfies

Quantum mechanical algorithms for the nonabelian hidden subgroup problem

\BPP \subseteq \BQP \subseteq \Ptime^{\SP}