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.
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.
Abstract Periodically driven quantum systems exhibit a diverse set of phenomena but are more challenging to simulate than their equilibrium counterparts. Here, we introduce the Quantum High-Frequency Floquet Simulation (QHiFFS) algorithm as a method to simulate fast-driven quantum systems on quantum hardware. Central to QHiFFS is the concept of a kick operator which transforms the system into a basis where the dynamics is governed by a time-independent effective Hamiltonian. This allows prior methods for time-independent simulation to be lifted to simulate Floquet systems. We use the periodically driven biaxial next-nearest neighbor Ising (BNNNI) model, a natural test bed for quantum frustrated magnetism and criticality, as a case study to illustrate our algorithm. We implemented a 20-qubit simulation of the driven two-dimensional BNNNI model on Quantinuum’s trapped ion quantum computer. Our error analysis shows that QHiFFS exhibits not only a cubic advantage in driving frequency ω but also a linear advantage in simulation time t compared to Trotterization.
Periodically driven quantum systems have emerged as one of the most versatile tools for quantum simulation, enabling the engineering of effective Hamiltonians, topological phases, and inherently nonequilibrium phenomena such as discrete time crystals. Yet fundamental challenges persist: from a spectral ambiguity that precludes a unique Floquet ground state to an inability to manipulate Floquet eigenstates beyond the adiabatic regime. This thesis demonstrates that counterdiabatic driving—a technique from adiabatic quantum control—provides a unifying framework to address these challenges. We first show that the periodic lab-frame Hamiltonian generates counterdiabatic driving for the Floquet eigenstates, revealing Floquet theory as an inverse counterdiabatic driving problem. Exploiting this equivalence, we develop geometric Floquet theory: a reformulation based on the parallel-transport gauge, which uniquely decomposes Floquet dynamics into a geometric contribution, encoded in Berry phases, and a dynamical contribution, generated by the average-energy operator. The average-energy operator provides an unambiguous sorting of the quasienergy spectrum, identifies a unique Floquet ground state at all drive frequencies, and serves as a sensitive diagnostic for heating and symmetry-breaking transitions. Inherently nonequilibrium phenomena—the 𝜋-quasienergy splitting of discrete time crystals and the anomalous edge modes of Floquet topological insulators—are traced to geometric phases. On the practical side, we exploit the intrinsic robustness of prethermal discrete time crystals for a highly frequency-selective DTC-based AC magnetic field sensing scheme and demonstrate it experimentally on a nuclear spin quantum simulator. We then generalize counterdiabatic driving to periodically driven systems. In particular, we derive a non-perturbative variational principle for the Floquet adiabatic gauge potential; this yields local, experimentally feasible control protocols for transitionless manipulation of Floquet states. We demonstrate this formalism on Floquet state preparation in an ultracold atom quantum simulator and many-body state transfer across a quantum phase transition, with significant improvements over unassisted protocols in both cases.
Analogue and digital Hamiltonian simulation represent two distinct approaches to making one quantum system replicate the physical properties of another, with consequences ranging from practical applications to complexity theory and quantum gravity. They are rooted in two distinct regimes of quantum dynamics: time-independent Hamiltonians for the former; fully controllable time-dependent quantum circuits in the latter. Time-periodic Hamiltonians represent an intermediate regime. In this work, we establish a theory of Floquet simulation where we use periodically driven Hamiltonians to synthesise time-independent ones.
We show that a set of interactions S can Floquet-simulate every Hamiltonian in the Lie algebra Lie(S), and consequently we give a complete and constructive characterisation of Floquet-universal Hamiltonians that are able to produce any target Hamiltonian. Notably, our construction uses only O(1) local interaction strengths and ratios thereof, avoiding the impractical multi-scale local interaction strengths often required in analogue simulation via time-independent Hamiltonians. Instead, the construction requires a range of driving frequencies which scales polynomially with system size for important classes such as k-local lattice Hamiltonians. This has immediate complexity-theoretic implications, including numerous BQP-completeness and QMA-hardness results for natural physical properties and quantities studied in Floquet physics.
Periodically driven systems have emerged as a useful technique to engineer the properties of quantum systems, and are in the process of being developed into a standard toolbox for quantum simulation. An outstanding challenge that leaves this toolbox incomplete is the manipulation of the states dressed by strong periodic drives. The state-of-the-art in Floquet control is the adiabatic change of parameters. Yet, this requires long protocols conflicting with the limited coherence times in experiments. To achieve fast control of nonequilibrium quantum matter, we generalize the notion of variational counterdiabatic driving away from equilibrium focusing on Floquet systems. We derive a nonperturbative variational principle to find local approximations to the adiabatic gauge potential for the effective Floquet Hamiltonian. It enables transitionless driving of Floquet eigenstates far away from the adiabatic regime. We discuss applications to two-level, Floquet band, and interacting periodically driven models. The developed technique allows us to capture nonperturbative photon resonances and obtain high-fidelity protocols that respect experimental limitations like the locality of the accessible control terms.
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.
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.