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
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.
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