On an implementation of the SolovayKitaev algorithm
Abstract
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 $polylog(1/\epsilon)$ with product length $polylog(1/\epsilon)$, 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 $d^2$ and requires memory of size exponential in $d^2$. In this paper we present methods which make the implementation of the existing algorithms easier. We present heuristic methods which make a timelength tradeoff 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.
 Publication:

arXiv eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:quantph/0606077
 Bibcode:
 2006quant.ph..6077N
 Keywords:

 Quantum Physics
 EPrint:
 10 pages, published on the 10th Rhine Workshop on Computer Algebra