The exact synthesis of 1- and 2-qubit Clifford+T circuits

TL;DR

This paper introduces an exact method for decomposing arbitrary 1- and 2-qubit operators with algebraic entries into minimal Clifford+T circuits, optimizing gate count and ancilla usage.

Contribution

It presents a new decomposition technique for certain algebraic quantum operators into Clifford+T gates with optimal bounds, extending previous methods.

Findings

Achieves O(k) gate complexity for the decomposition.

Uses at most one ancilla qubit in the process.

Builds on and improves the Giles and Selinger approach.

Abstract

We describe a new method for the decomposition of an arbitrary $n$ qubit operator with entries in $\mathbb{Z}[i,\frac{1}{\sqrt{2}}]$, i.e., of the form $(a+b\sqrt{2}+i(c+d\sqrt{2}))/{\sqrt{2}^{k}}$, into Clifford+$T$ operators where $n\le 2$. This method achieves a bound of $O(k)$ gates using at most one ancilla using decomposition into $1$- and $2$-level matrices which was first proposed by Giles and Selinger.