On an implementation of the Solovay-Kitaev algorithm

In quantum computation we are given a finite set of gates and we have to perform a desired operation as a product of them. The corresponding computational problem is approximating an arbitrary unitary as a product in a topological generating set of SU(d). The problem is known to be solvable in time polylog(1/ǫ) with product length polylog(1/ǫ), where the implicit constants depend on the given generators. The existing algorithms solve the problem but they need a very slow and space consuming preparatory stage. This stage runs in time exponential in d and requires memory of size exponential in d. In this paper we present methods which make the implementation of the existing algorithms easier. We present heuristic methods which make a time-length trade-off in the preparatory step. We decrease the running time and the used memory to polynomial in d but the length of the products approximating the desired operations will increase (by a factor which depends on d). We also present a simple method which can be used for decomposing a unitary into a product of group commutators for 2 < d < 256, which is an important part of the existing algorithm.