Characterizing classes of functions computable by quantum parallelism

Abstract

Computation by quantum parallelism involves associating a quantum state v(f) to each function f : Z m → Z n . v(f) is formed from superpositions of states labelled by the values of f , the standard choice being an equally weighted superposition of all the values of f . A joint property G(f) of the values f (0),..., f ( m –1) is called computable by quantum parallelism (QPC) if there is an observable g which will reveal the value G(f) , with non-zero probability, when applied to v(f) , and will never show a false value. It is shown that the problem of deciding which G s are QPC can be formulated entirely in terms of the linear relations which exist among the v(f) s. In the case of f : Z m → Z 2 , G :( Z 2 ) m → Z 2 we explicitly describe all properties which are QPC and show that this includes only m 2 + m + 2 of the 2 2 m such properties. By using a suitable nonstandard definition for v(f) this number can be increased to 2 2 m — 2 m+1 (for m > 1).