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, 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 defines two different kinds of eigenstates, the query complexity of the algorithms achieves the near-optimal scaling in allowable 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.
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, 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 defines two different kinds of eigenstates, the query complexity of the algorithms achieves the near-optimal scaling in allowable 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.
The implementation of time-evolution operators U(t) , called Hamiltonian simulation, is one of the most promising usage of quantum computers. For time-independent Hamiltonians, qubitization has recently established efficient realization of time-evolution U(t)=e−iHt , with achieving the optimal computational resource both in time t and an allowable error ε . In contrast, those for time-dependent systems require larger cost due to the difficulty of handling time-dependency. In this paper, we establish optimal/nearly-optimal Hamiltonian simulation for generic time-dependent systems with time-periodicity, known as Floquet systems. By using a so-called Floquet-Hilbert space equipped with auxiliary states labeling Fourier indices, we develop a way to certainly obtain the target time-evolved state without relying on either time-ordered product or Dyson-series expansion. Consequently, the query complexity, which measures the cost for implementing the time-evolution, has optimal and nearly-optimal dependency respectively in time t and inverse error ε , and becomes sufficiently close to that of qubitization. Thus, our protocol tells us that, among generic time-dependent systems, time-periodic systems provides a class accessible as efficiently as time-independent systems despite the existence of time-dependency. As we also provide applications to simulation of nonequilibrium phenomena and adiabatic state preparation, our results will shed light on nonequilibrium phenomena in condensed matter physics and quantum chemistry, and quantum tasks yielding time-dependency in quantum computation.
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 ε, 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 ε, and simple oracles as well as the optimal algorithm for time-independent cases.
Simulating Hamiltonian dynamics is one of the most fundamental and significant tasks for characterizing 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 singular value transformation with the optimal query complexity in time t and desirable accuracy $\ensuremath{\varepsilon}$, 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 multiperiodic time-dependent Hamiltonians with optimal and nearly optimal query complexity both in time t and desirable accuracy $\ensuremath{\varepsilon}$, and simple oracles as well as the optimal algorithm for time-independent cases.
We study driven and open quantum systems described by a time-periodic Lindblad master equation. In closed systems, the stroboscopic dynamics can always be described by an effective time-independent Floquet Hamiltonian; this idea is the basis of Floquet engineering. However, in the presence of dissipation, the existence of an effective time-independent Floquet Lindbladian is not guaranteed due to the non-unitary nature of the evolution. Using Floquet theory, we construct a well-defined time-independent Floquet Lindbladian in an extended Sambe-Liouville space, transforming the initial time-dependent problem to a static and non-Hermitian eigenvalue problem. For harmonic driving, we introduce a matrix continued fraction method to nonperturbatively resum multiphoton processes and construct an effective Floquet Lindbladian acting only on the physical Liouville space. Compared to other high-frequency expansions, this method has the advantage of providing the whole infinite series expansion at once. Using a resolvent formalism, we show how to obtain a spectral Floquet representation of correlation functions of an open quantum system. As an application, we consider a dissipating two-level system in a linearly polarized field and calculate its resonance fluorescence spectrum. Furthermore, we consider a parametrically driven quantum dot with pump and loss for which we calculate its spectral function and current-voltage characteristics.
Floquet engineering—the periodic driving of quantum systems—offers a powerful pathway to realize exotic nonequilibrium phases, including topological Floquet insulators and discrete time crystals. However, computing the quasienergy band structure of interacting, many-body Floquet systems remains a formidable challenge for classical computers due to the exponential growth of the Hilbert space and the difficulty of diagonalizing the non-Hermitian effective Floquet Hamiltonian. In this work, we introduce and experimentally demonstrate a variational quantum algorithm (VQA) tailored specifically for the near-term computation of Floquet quasienergy bands. Unlike standard variational eigensolvers that minimize static energy expectations, our approach employs a parameterized quantum circuit to prepare variational trial states and directly extracts the quasienergy spectrum via a Hadamard-test-based cost function that measures the complex overlap with the single-period Floquet unitary propagator. By scanning the conserved crystal momentum, we reconstruct the full quasienergy dispersion for both one- and two-dimensional lattice models. We benchmark the algorithm against exact diagonalization for system sizes up to N=8N=8 qubits and subsequently deploy it on superconducting transmon processors, systematically characterizing the impact of decoherence, gate infidelity, and readout errors on the resolved band structure. To mitigate hardware noise, we integrate zero-noise extrapolation and measurement-error mitigation, recovering quasienergy features with fidelities exceeding 90% for shallow circuit depths. Our results establish that variational phase-estimation protocols offer a scalable, hardware-efficient route to exploring driven quantum matter, providing a practical toolkit for simulating Floquet topological phenomena on current-generation quantum devices well before the advent of fault-tolerant quantum computation.
Periodically driven quantum systems exhibit many fascinating phenomena absent in equilibrium systems, but their simulation is more challenging than that of static systems. Consequently, quantum simulation of these systems offers greater opportunity for achieving quantum advantage. To build the foundation for simulating time-periodic Hamiltonians, we utilize the Floquet-Hilbert space formalism to transform the Hamiltonian into a time-independent form and provide its Pauli decomposition. Within this formalism, we develop a hybrid quantum-classical algorithm, Floquet-ADAPT-VQE, that produces three key quantities that govern periodically driven quantum dynamics: the quasienergy, the associated Floquet eigenstates, and expectation values of observables with respect to these eigenstates. The latter is achieved by applying a fixed-depth quantum circuit to an initial Floquet eigenstate—an efficient construction made possible by the Floquet-Hilbert space formalism. We also identify a suitable auxiliary initial state whose squared Floquet energy is independent of the number of auxiliary qubits as well as the driving frequency, which leads to better convergence with fewer ADAPT iterations. We demonstrate our algorithm by performing numerical simulations on a periodically driven XYZ model with a magnetic field. We also explore potential applications of our algorithm for studying various non-equilibrium phenomena in periodically driven systems.