Representation of Quantum Circuits with Clifford and $\pi/8$ Gates

TL;DR

This paper introduces a normal form for one-qubit quantum circuits over the Clifford and T gates, enabling easier circuit comparison and counting the number of unitaries with limited T-gates.

Contribution

It presents a unique normal form for circuits over the standard basis, facilitating circuit equivalence checks and enumeration of unitaries with bounded T-gates.

Findings

Every circuit can be transformed into the normal form.

Two normal form circuits compute the same unitary iff they are identical.

Number of unitaries with at most n T-gates is exactly 192*(3*2^n - 2).

Abstract

In this paper, we introduce the notion of a normal form of one qubit quantum circuits over the basis $\{H, P, T\}$, where $H$, $P$ and $T$ denote the Hadamard, Phase and $\pi/8$ gates, respectively. This basis is known as the {\it standard set} and its universality has been shown by Boykin et al. [FOCS '99]. Our normal form has several nice properties: (i) Every circuit over this basis can easily be transformed into a normal form, and (ii) Every two normal form circuits compute same unitary matrix if and only if both circuits are identical. We also show that the number of unitary operations that can be represented by a circuit over this basis that contains at most $n$ $T$-gates is exactly $192 \cdot (3 \cdot 2^n - 2)$.