Synthesis of CNOT-Dihedral circuits with optimal number of two qubit gates

Shelly Garion1 and Andrew W. Cross2

1IBM Quantum, IBM Research Haifa, Haifa University Campus, Mount Carmel, Haifa, 3498825, Israel
2IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA

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In this note we present explicit canonical forms for all the elements in the two-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 algorithm 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.

CNOT-Dihedral circuits are certain quantum circuits that are generated using the quantum gates X, T and controlled-X (CX). These circuits can be efficiently simulated in polynomial time using a classical computer and have applications for quantum error correction codes and for estimating quantum gate errors using a benchmarking process. It is therefore important to efficiently synthesize CNOT-Dihedral circuits using a minimal number of physical basic gates, in particular, two-qubit gates.
In this note we characterize all the two-qubit CNOT-Dihedral circuits, allowing a synthesis of such circuits with optimal number of certain two-qubit gates, and provide an algorithm to successively construct n-qubit CNOT-Dihedral circuits, asserting an optimal number of CX gates.

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Cited by

[1] Shelly Garion, Naoki Kanazawa, Haggai Landa, David C. McKay, Sarah Sheldon, Andrew W. Cross, and Christopher J. Wood, "Experimental implementation of non-Clifford interleaved randomized benchmarking with a controlled-S gate", arXiv:2007.08532.

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