  Non-autonomous dynamical systems appear in a very wide range of interesting
applications, both in classical and quantum dynamics, where in the latter case
it corresponds to having a time-dependent Hamiltonian. However, the quantum
simulation of these systems often needs to appeal to rather complicated
procedures involving the Dyson series, considerations of time-ordering,
requirement of time steps to be discrete and/or requiring multiple measurements
and postselection. These procedures are generally much more complicated than
the quantum simulation of time-independent Hamiltonians. Here we propose an
alternative formalism that turns any non-autonomous unitary dynamical system
into an autonomous unitary system, i.e., quantum system with a time-independent
Hamiltonian, in one higher dimension, while keeping time continuous. This makes
the simulation with time-dependent Hamiltonians not much more difficult than
that of time-independent Hamiltonians, and can also be framed in terms of an
analogue quantum system evolving continuously in time. We show how our new
quantum protocol for time-dependent Hamiltonians can be performed in a
resource-efficient way and without measurements, and can be made possible on
either continuous-variable, qubit or hybrid systems. Combined with a technique
called Schrodingerisation, this dilation technique can be applied to the
quantum simulation of any linear ODEs and PDEs, and nonlinear ODEs and certain
nonlinear PDEs, with time-dependent coefficients.
