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Time-periodic (Floquet) systems are one of the most interestingnonequilibrium systems. As the computation of energy eigenvalues andeigenstates of time-independent Hamiltonians is a central problem in bothclassical and quantum computation, quasienergy and Floquet eigenstates are theimportant targets. However, their computation has difficulty of timedependence; the problem can be mapped to a time-independent eigenvalue problemby the Sambe space formalism, but it instead requires additional infinitedimensional space and seems to yield higher computational cost than thetime-independent cases. It is still unclear whether they can be computed withguaranteed accuracy as efficiently as the time-independent cases. We addressthis issue by rigorously deriving the cutoff of the Sambe space to achieve thedesired accuracy and organizing quantum algorithms for computing quasienergyand Floquet eigenstates based on the cutoff. The quantum algorithms returnquasienergy and Floquet eigenstates with guaranteed accuracy like Quantum PhaseEstimation (QPE), which is the optimal algorithm for outputting energyeigenvalues and eigenstates of time-independent Hamiltonians. While the timeperiodicity provides the additional dimension for the Sambe space and ramifiesthe eigenstates, the query complexity of the algorithms achieves thenear-optimal scaling in allwable errors. In addition, as a by-product of thesealgorithms, we also organize a quantum algorithm for Floquet eigenstatepreparation, in which a preferred gapped Floquet eigenstate can bedeterministically implemented with nearly optimal query complexity in the gap.These results show that, despite the difficulty of time-dependence, quasienergyand Floquet eigenstates can be computed almost as efficiently astime-independent cases, shedding light on the accurate and fast simulation ofnonequilibrium systems on quantum computers.

Nearly optimal quasienergy estimation and eigenstate preparation of time-periodic Hamiltonians by Sambe space formalism