Clifford-cyclotomic Circuits

Abstract

Circuit design is an important aspect of quantum computation theory, and the question of constructing a circuit that exactly represents a given operator is prominent in the research area. Let n be a positive integer divisible by 8. The Clifford-cyclotomic gate set \mathcal{G}_n consists of the Clifford gates, together with a z-rotation of order n. It is easy to show that, if a circuit over \mathcal{G}_n represents a unitary matrix U, then the entries of U must lie in \mathcal{R}_n, the smallest subring of \mathbb{C} containing 1/2 and \mathrm{exp}(2\pi i/n). The converse implication, that every unitary U with entries in \mathcal{R}_n can be represented by a circuit over \mathcal{G}_n, is harder to show, but it was recently proved to be true when n=2^k. In that case, k-2 ancillas suffice to synthesize a circuit for U, which is known to be minimal for k=3, but not for larger values of k. In the present thesis, we make two contributions to the theory of Clifford-cyclotomic circuits. Firstly, we improve the existing synthesis algorithm by showing that, when n=2^k and k\geq 4, only k-3 ancillas are needed to synthesize a circuit for U, which is minimal for k=4. Secondly, we extend the existing synthesis algorithm to the case of n=3\cdot 2^k with k\geq 3.