  Simulating Hamiltonian dynamics is one of the most fundamental and
significant tasks for characterising quantum materials. Recently, a series of
quantum algorithms employing block-encoding of Hamiltonians have succeeded in
providing efficient simulation of time-evolution operators on quantum
computers. While time-independent Hamiltonians can be simulated by the quantum
eigenvalue transformation (QET) or quantum singularvalue transformation with
the optimal query complexity in time $t$ and desirable accuracy $\varepsilon$,
generic time-dependent Hamiltonians face at larger query complexity and more
complicated oracles due to the difficulty of handling time-dependency. In this
paper, we establish a QET-based approach for simulating time-dependent
Hamiltonians with multiple time-periodicity. Such time-dependent Hamiltonians
involve a variety of nonequilibrium systems such as time-periodic systems
(Floquet systems) and time-quasiperiodic systems. Overcoming the difficulty of
time-dependency, our protocol can simulate the dynamics under multi-periodic
time-dependent Hamiltonians with optimal/nearly-optimal query complexity both
in time $t$ and desirable accuracy $\varepsilon$, and simple oracles as well as
the optimal algorithm for time-independent cases.
