Necessary and Sufficient Condition for Quantum Computing

Abstract

A necessary and sufficient condition for quantum computing performed with, for example, the Deutsch-Jozsa algorithm or the Bernstein-Vazirani algorithm, has theoretically been investigated. Assume a 2N qubit-quantum computing which starts with the state | N 0, 0,..., 0, 1 | N 1, 1,..., 1 as follows: Uf | 0, 0,..., 0, 1 | 1, 1,..., 1 = | 0, 0,..., 0, 1 | f (0, 0,..., 0, 1) . Surprisingly the relation f (x) = f (-x) is the necessary and sufficient condition of holding this fundamental relation if local unitary operations can be used.