Exponential improvement in precision for simulating sparse Hamiltonians

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D. W. Berry, A. M. Childs, R. Cleve, R. Kothari, R. D. Somma
Proceedings of the 46th ACM Symposium on Theory of Computing (STOC 2014), 283-292 (2014)

Simulation of quantum mechanical systems is a major potential application of quantum computers. Indeed, the problem of simulating Hamiltonian dynamics was the original motivation for the idea of quantum computation.

In their work, Berry and co-workers provide a quantum algorithm for simulating the dynamics of sparse Hamiltonians with complexity sublogarithmic in the inverse error, an exponential improvement over previous methods. Unlike previous approaches based on product formulas, the query complexity is independent of the number of qubits acted on, and for time-varying Hamiltonians, the gate complexity is logarithmic in the norm of the derivative of the Hamiltonian. The algorithm is based on a significantly improved simulation of the continuous, and fractional, query models using discrete quantum queries, showing that the former models are not much more powerful than the discrete model even for very small error. Finally, they prove that the algorithm is optimal as a function of the error.