Here trp is the partial trace over the first variable and S is the swap operator.S is a sparse matrix and its elements are efficiently computable,so e-isA can be performed efficiently-14.Repeated application of (1)with n copies of p allows one to construct e-ipmAePA.Comparison with the Suzuki-Trotter theory of quantum simulation-shows that to simulate e to accuracy e requires n=O(t2ep-)O(t2e-)steps,where t=nAt and is the sup norm.Thus,simply performing repeated infinitesimal swap operations on po allows us to construct the unitary operator