Discrete simulations of continuous - time query algorithms that are efficient with respect to queries , gates and space ∗

The continuous-time query model [3] can be thought of as a variant of the stardard query model, but where arbitrarily small λ-fractional queries to the data x1x2 . . . xN ∈ {0, 1} , of the form |j⟩ 7→ ej |j⟩, can be made at cost only λ. In the limit as λ → 0, such algorithms become continuous-time Hamiltonian evolution processes. We show that any continuous-time quantum query algorithm whose total query time is T and whose driving Hamiltonian is implementable with G 1and 2-qubit gates (in a sense defined below) can be simulated by a discrete-query quantum algorithm using the following resources: