A Framework for Approximating Qubit Unitaries

We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $varepsilon$-approximations using circuits of length $O(log(1/varepsilon))$, which is asymptotically optimal. The algorithm achieves the same quality of approximation as previously-known algorithms for Clifford+T [arXiv:1212.6253], V-basis [arXiv:1303.1411] and Clifford+$pi/12$ [arXiv:1409.3552], running on average in time polynomial in $O(log(1/varepsilon))$ (conditional on a number-theoretic conjecture). Ours is the first such algorithm that works for a wide range of gate sets and provides insight into what should constitute a good gate set for a fault-tolerant quantum computer.