  Compared with time independent Hamiltonians, the dynamics of generic quantum
Hamiltonians $H(t)$ are complicated by the presence of time ordering in the
evolution operator. In the context of digital quantum simulation, this
difficulty prevents a direct adaptation of time independent simulation
algorithms for time dependent simulation. However, there exists a framework
within the theory of dynamical systems which eliminates time ordering by adding
a "clock" degree of freedom. In this work, we provide a computational
framework, based on this reduction, for encoding time dependent dynamics as
time independent systems. As a result, we make two advances in digital
Hamiltonian simulation. First, we create a time dependent simulation algorithm
based on performing qubitization on the augmented clock system, and in doing
so, provide the first qubitization-based approach to time dependent
Hamiltonians that goes beyond Trotterization of the ordered exponential.
Second, we define a natural generalization of multiproduct formulas for
time-ordered exponentials, then propose and analyze an algorithm based on these
formulas. Unlike other algorithms of similar accuracy, the multiproduct
approach achieves commutator scaling, meaning that this method outperforms
existing methods for physically-local time dependent Hamiltonians. Our work
reduces the disparity between time dependent and time independent simulation
and indicates a step towards optimal quantum simulation of time dependent
Hamiltonians.
