We prove that a unitary matrix has an exact representation over the Clifford+$T$ gate set with local ancillas if and only if its entries are in the ring $\mathbb{Z}[\frac{1}{\sqrt{2}},i]$. Moreover, we show that one ancilla always suffices. These facts were conjectured by Kliuchnikov, Maslov, and Mosca. We obtain an algorithm for synthesizing a exact Clifford+$T$ circuit from any such $n$-qubit operator. We also characterize the Clifford+$T$ operators that can be represented without ancillas.

• Received 4 December 2012

DOI:https://doi.org/10.1103/PhysRevA.87.032332