  Time-periodic (Floquet) systems are one of the most interesting
nonequilibrium systems. As the computation of energy eigenvalues and
eigenstates of time-independent Hamiltonians is a central problem in both
classical and quantum computation, quasienergy and Floquet eigenstates are the
important targets. However, their computation has difficulty of time
dependence; the problem can be mapped to a time-independent eigenvalue problem
by the Sambe space formalism, but it instead requires additional infinite
dimensional space and seems to yield higher computational cost than the
time-independent cases. It is still unclear whether they can be computed with
guaranteed accuracy as efficiently as the time-independent cases. We address
this issue by rigorously deriving the cutoff of the Sambe space to achieve the
desired accuracy and organizing quantum algorithms for computing quasienergy
and Floquet eigenstates based on the cutoff. The quantum algorithms return
quasienergy and Floquet eigenstates with guaranteed accuracy like Quantum Phase
Estimation (QPE), which is the optimal algorithm for outputting energy
eigenvalues and eigenstates of time-independent Hamiltonians. While the time
periodicity provides the additional dimension for the Sambe space and ramifies
the eigenstates, the query complexity of the algorithms achieves the
near-optimal scaling in allwable errors. In addition, as a by-product of these
algorithms, we also organize a quantum algorithm for Floquet eigenstate
preparation, in which a preferred gapped Floquet eigenstate can be
deterministically implemented with nearly optimal query complexity in the gap.
These results show that, despite the difficulty of time-dependence, quasienergy
and Floquet eigenstates can be computed almost as efficiently as
time-independent cases, shedding light on the accurate and fast simulation of
nonequilibrium systems on quantum computers.
