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We present an algorithm for building a circuit that approximates single qubitunitaries with precision {\epsilon} using O(log(1/{\epsilon})) Clifford and Tgates and employing up to two ancillary qubits. The algorithm for computing ourapproximating circuit requires an average of O(log^2(1/{\epsilon})loglog(1/{\epsilon})) operations. We prove that the number of gates in our circuitsaturates the lower bound on the number of gates required in the scenario whena constant number of ancillae are supplied, and as such, our circuits areasymptotically optimal. This results in significant improvement over thecurrent state of the art for finding an approximation of a unitary, includingthe Solovay-Kitaev algorithm that requires O(log^{3+{\delta}}(1/{\epsilon}))gates and does not use ancillae and the phase kickback approach that requiresO(log^2(1/{\epsilon})log log(1/{\epsilon})) gates, but usesO(log^2(1/{\epsilon})) ancillae.

Asymptotically optimal approximation of single qubit unitaries by Clifford and T circuits using a constant number of ancillary qubits