Exact synthesis of multiqubit Clifford plus T circuits

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 Z[1/root 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. DOI: 10.1103/PhysRevA.87.032332