Asymptotically Optimal Approximation of Single Qubit Unitaries by Clifford and $T$ Circuits Using a Constant Number of Ancillary Qubits

Decomposing unitaries into a sequence of elementary operations is at the core of quantum computing. Information theoretic arguments show that approximating a random unitary with precision $\epsilon$ requires $\Omega \mathbf{(}\log (1/\epsilon )\mathbf{)}$ gates. Prior to our work, the state of the art in approximating a single qubit unitary included the Solovay-Kitaev algorithm that requires $O\mathbf{(}{log}^{3+\delta }(1/\epsilon )\mathbf{)}$ gates and does not use ancillae and the phase kickback approach that requires $O\mathbf{(}{log}^2(1/\epsilon )\log \log (1/\epsilon )\mathbf{)}$ gates but uses $O\mathbf{(}{log}^2(1/\epsilon )\mathbf{)}$ ancillae. Both algorithms feature upper bounds that are far from the information theoretic lower bound. In this Letter, we report an algorithm that saturates the lower bound, and as such it guarantees asymptotic optimality. In particular, we present an algorithm for building a circuit that approximates single qubit unitaries with precision $\epsilon$ using $O\mathbf{(}\log (1/\epsilon )\mathbf{)}$ Clifford and $T$ gates and employing up to two ancillary qubits. We connect the unitary approximation problem to the problem of constructing solutions corresponding to Lagrange’s four-square theorem, and thereby develop an algorithm for computing an approximating circuit using an average of $O\mathbf{(}{log}^2(1/\epsilon )\log \log (1/\epsilon )\mathbf{)}$ operations with integers.