Optimal ancilla-free Clifford+V approximation of z-rotations
Neil J. Ross · Quantum Information and Computation · 2015
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.