Hamiltonian simulation with complexity polylogarithmic in the error

Quantum algorithms for the simulation of quantum systems described by a Hamiltonian provide an exponential improvement in speed over classical algorithms when one considers scaling of the complexity in terms of the dimension. However, one key drawback is that the scaling in terms of the allowable error, ε, is relatively poor. Here we provide a new quantum algorithm whose scaling with respect to the allowable error is exponentially smaller than previous algorithms. That is, the complexity is polylogarithmic in 1/ε, rather than polynomial. The algorithm’s scaling with respect to other parameters—such as dimension and evolution time—also compares well with previous algorithms.