Geometric quantum computation

We describe in detail a general strategy for implementing a conditional geo metric phase between two spins. Combined with single-spin operations, this simple operation is a universal gate for quantum computation, in that any u nitary transformation can be implemented with arbitrary precision using onl y single-spin operations and conditional phase shifts. Thus quantum geometr ical phases can form the basis of any quantum computation. Moreover, as the induced conditional phase depends only on the geometry of the paths execut ed by the spins it is resilient to certain types of errors and offers the p otential of a naturally fault-tolerant way of performing quantum computatio n.