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The simplified Toffoli gate implementation by Margolus is optimal

Guang Song·Andreas Klappenecker
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Abstract

Unitary operations are expressed in the quantum circuit model as a finite sequence of elementary gates, such as controlled-not gates and single qubit gates. We prove that the simplified Toffoli gate by Margolus, which coincides with the Toffoli gate up to a single change of sign, cannot be realized with less than three controlled-not gates. If the circuit is implemented with three controlled-not gates, then at least four additional single qubit gates are necessary. This proves that the implementation suggested by Margolus is optimal.

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Guang Song, Andreas Klappenecker. 2003-12-31. The simplified Toffoli gate implementation by Margolus is optimal. https://arxiv.org/abs/quant-ph/0312225

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