We give a query-optimal algorithm for simulating a general Lipschitz-continuous time-dependent Hamiltonian. Its query complexity matches the known lower bound for time-independent Hamiltonian simulation, showing that time dependence incurs no asymptotic query overhead. The main ingredient is a one-query transducer that implements the target evolution given a catalyst state. A weighted combination of circuits that reuse the transducer different numbers of times removes the need to prepare the catalyst, with the error from omitting the catalyst decaying factorially and thereby yielding the optimal dependence on precision. We further show that the same algorithm admits a gate-efficient implementation that retains its optimal query complexity, with gate complexity linear in the query complexity up to logarithmic factors.

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