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

TL;DR

This paper provides explicit canonical forms for two-qubit CNOT-Dihedral circuits with minimal controlled gates and introduces an algorithm for constructing n-qubit CNOT-Dihedral groups with optimal gate counts, aiding quantum circuit optimization.

Contribution

It introduces explicit canonical forms for two-qubit CNOT-Dihedral circuits and an algorithm for optimal n-qubit circuit construction, advancing quantum gate efficiency.

Findings

Canonical forms for two-qubit CNOT-Dihedral circuits with minimal gates.

Algorithm for constructing n-qubit CNOT-Dihedral groups with optimal controlled-X gates.

Potential applications in quantum circuit optimization and error estimation.

Abstract

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.