The Pauli matrices are a set of three $2\times 2$ complex Hermitian unitary matrices. In this article, we investigate the relationships between certain roots of the Pauli matrices and how gates implementing those roots are used in quantum circuits. Techniques for simplifying such circuits are given. In particular, we show how those techniques can be used to find a circuit of $\text{Clifford}+T$ gates starting from a circuit composed of gates from the well-studied NOT, CNOT, V library.

• Received 13 August 2013

DOI:https://doi.org/10.1103/PhysRevA.88.042322