A Framework for Approximating Qubit Unitaries

Abstract

We present an algorithm for efficiently approximating of qubit unitaries over gate sets derived from totally definite quaternion algebras. It achieves $\epsilon$-approximations using circuits of length $O(\log (1/\epsilon ))$, 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 $O(\log (1/\epsilon ))$ (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.