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In this work we provide a new approach for approximating an ordered operatorexponential using an ordinary operator exponential that acts on the Hilbertspace of the simulation as well as a finite-dimensional clock register. Thisapproach allows us to translate results for simulating time-independent systemsto the time-dependent case. Our result solves two open problems in simulation.It first provides a rigorous way of using discrete time-displacement operatorsto generate time-dependent product and multiproduct formulas. Second, itprovides a way to allow qubitization to be employed for certain time-dependentformulas. We show that as the number of qubits used for the clock grows, theerror in the ordered operator exponential vanishes, as well as the entanglementbetween the clock register and the register of the state being acted upon. Asan application, we provide a new family of multiproduct formulas (MPFs) fortime-dependent Hamiltonians that yield both commutator scaling andpoly-logarithmic error scaling. This in turn, means that this methodoutperforms existing methods for simulating physically-local, time-dependentHamiltonians. Finally, we apply the formalism to show how qubitization can begeneralized to the time-dependent case and show that such a translation can bepractical if the second derivative of the Hamiltonian is sufficiently small.

Time-dependent Hamiltonian Simulation Using Discrete Clock Constructions