We describe a new efficient algorithm to approximate z-rotations by
ancilla-free Clifford+V circuits, up to a given precision epsilon. Our
algorithm is optimal in the presence of an oracle for integer factoring: it
outputs the shortest Clifford+V circuit solving the given problem instance. In
the absence of such an oracle, our algorithm is still near-optimal, producing
circuits of V-count m + O(log(log(1/epsilon))), where m is the V-count of the
third-to-optimal solution. A restricted version of the algorithm approximates
z-rotations in the Pauli+V gate set. Our method is based on previous work by
the author and Selinger on the optimal ancilla-free approximation of
z-rotations using Clifford+T gates and on previous work by Bocharov, Gurevich,
and Svore on the asymptotically optimal ancilla-free approximation of
z-rotations using Clifford+V gates.
