Elementary gates for quantum computation

Abstract

We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values \((x,y)\) to \((x,x \oplus y)\)) is universal in the sense that all unitary operations on arbitrarily many bits \(n\) (U(\(2^n\))) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two-and three-bit quantum gates, the asymptotic number required for \(n\)-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary \(n\)-bit unitary operations.