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On the Structure of the CNOT-Dihedral Group
Shelly Garion, Andrew W. Cross·June 22, 2020
PhysicsMathematics
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Abstract
In this note we present explicit canonical forms for all the elements in the $2$-qubit CNOT-Dihedral group, with minimal numbers of controlled-$S$ ($CS$) and controlled-$X$ ($CX$) gates, using the generating set of quantum gates $[X, T, CX, CS]$. We provide an efficient algorithm, with a sharp bound, to successively construct the $n$-qubit CNOT-Dihedral group, asserting an optimal number of controlled-$X$ ($CX$) gates. These results are needed to estimate gate errors via non-Clifford randomized benchmarking and may have further applications to circuit optimization over fault tolerant gate sets.