Contributions to the Theory of Clifford-Cyclotomic Circuits

Let n be a positive integer divisible by 8.The Clifford-cyclotomic gate set 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 G n represents a unitary matrix U, then the entries of U must lie in R n , the smallest subring of C containing 1/2 and exp(2i/n).The converse implication, that every unitary U with entries in R n can be represented by a circuit over 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 paper, 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 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 2 k with k 3.