Quantum complexity theory
y z
Ethan Bernstein Umesh Vazirani
September
Abstract
In this pap er we study quantum computation from a complexity theoretic viewp oint Our rst result
is the existence of an e cient universal quantum Turing Machine in Deutsch s mo del of a quantum Turing
Machine This construction is substantially more complicated than the corresp onding construction
for classical Turing Machines in fact even simple primitives such as lo oping branching and comp osition
are not straightforward in the context of quantum Turing Machines We establish how these familiar
primitives can b e implemented and also introduce some new purely quantum mechanical primitives
such as changing the computational basis and carrying out an arbitrary unitary transformation of
p olynomially b ounded dimension
We also consider the precision to which the transition amplitudes of a quantum Turing Machine need
to b e sp eci ed We prove that O log T bits of precision su ce to supp ort a T step computation This
justi es the claim that that the quantum Turing Machine mo del should b e regarded as a discrete mo del
of computation and not an analog one
We give the rst formal evidence that quantum Turing Machines violate the mo dern complexity
theoretic formulation of the Church Turing thesis We show the existence of a problem relative to
an oracle that can b e solved in p olynomial time on a quantum Turing Machine but requires sup er
p olynomial time on a b ounded error probabilistic Turing Machine and thus not in the class BPP
The class BQP of languages that are e ciently decidable with small error probability