Matthew Amy, Dmitri Maslov, Michele Mosca, Martin Roetteler
Many quantum computing algorithms are described at a high level of abstraction, using logical operations that can be quite complex. However, on a realistic scalable quantum computer, these logical operations would have to be compiled down to a small set of elementary gates which then have to be performed in a fault-tolerant manner. We present an algorithm for computing depth-optimal decompositions of logical operations, leveraging a {\it meet-in-the-middle} technique to provide a significant speed-up over simple brute force algorithms. As an illustration of our method, we present a decomposition of the Toffoli gate over the set of Clifford and $T$-gates. This elementary gate set arises, e.g., from fault-tolerant computing over the Steane code or the surface code. Our decomposition achieves a total $T$-depth of 3, thereby providing a 40% reduction over the previously best known decomposition for the Toffoli gate.
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http://arxiv.org/abs/1206.0758
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