Optimal realizations of controlled unitary gates

Authors:
Guang Song;Andreas Klappenecker
Affiliations:
Department of Computer Science, Texas A&M University, College Station, TX;Department of Computer Science, Texas A&M University, College Station, TX
Venue:
Quantum Information & Computation
Year:
2003

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Abstract

The controlled-not gate and the single qubit gates are considered elementary gates in quantum computing. It is natural to ask how many such elementary gates are needed to implement more elaborate gates or circuits. Recall that a controlled-U gate can be realized with two controlled-not gates and four single qubit gates. We prove that this implementation is optimal if and only if the matrix U satisfies the conditions tr U ≠ 0, tr(UX) ≠ 0, and det U ≠ 1. We also derive optimal implementations in the remaining non-generic cases.