Quantum complexity theory

Abstract

In this dissertation we study quantum computation from a complexity theoretic viewpoint. Our first result is the existence of an efficient universal quantum Turing Machine in Deutsch's model of a quantum Turing Machine. This construction is substantially more complicated than the corresponding construction for classical Turing Machines--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 also 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 O(log T) bits of precision suffice to support a T 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 evidence indicating that quantum Turing Machines are more powerful than classical probabilistic Turing Machines. 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 super-polynomial time on a bounded-error probabilistic Turing Machine; and thus not in the class BPP. In fact, we show that this problem cannot be solved in MA relative to the same oracle, thus showing that even non-determinism together with randomness is not sufficient to solve the problem in poly-nomial time. The class BQP, of languages that are efficiently decidable (with small error-probability) on a quantum Turing Machine, satisfies: BPP $\subseteq$ BQP $\subseteq$ P$\sp{\sharp P}$. Therefore there is no possibility of giving a mathematical proof that quantum Turing Machines are more powerful than classical probabilistic Turing Machines (in the unrelativized setting) unless there is a major breakthrough in complexity theory. We also give evidence suggesting that quantum Turing Machines cannot efficiently solve all of NP. Specifically, we prove that relative to an oracle chosen uniformly at random, with probability 1, the class NP cannot be solved on a quantum Turing machine in time $o(2\sp{n/2}).$