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Implementing Variational Quantum Algorithms to Compute Floquet Quasienergy Bands on Near-Term Hardware

Dane Kalorii

Published Jul 16, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.14293/pr2199.004115.v1

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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.
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.
This work addresses the critical challenge of excited-state preparation for semiconductor band structure calculations. We introduce a variational adiabatic eigensolver (VAE) protocol that combines adiabatic evolution with variational optimization to prepare high-fidelity eigenstates on noisy intermediate-scale quantum (NISQ) devices. Applying a momentum-space truncation, we accurately compute the electronic band structure of silicon -- an idealized infinite periodic system -- using only a modest number of qubits. Our approach employs multi-qubit parameterized circuits and a phase-based loss function, overcoming limitations of conventional methods. These limitations include the circuit-construction difficulty in traditional adiabatic approaches and the reduced accuracy of variational quantum eigensolvers for excited states. Through rigorous numerical simulation and experimental implementation on a superconducting quantum processor, we successfully prepare silicon's valence-band and conduction-band eigenstates. Single-shot readout yields state fidelities exceeding 96%, and the measured energy expectations agree with theoretical band energies within 0.5 eV. Further refinement via single-frequency oscillation fitting reduces the energy deviation to below 0.01 eV. This framework provides a robust and practical pathway for precisely determining electronic structures in quantum materials.
Many-body phenomena far from equilibrium present challenges beyond reach by classical computational resources. Digital quantum computers provide a possible way forward but noise limits their use in the near term. We propose a scheme to simulate and characterize many-body Floquet systems hosting a rich variety of phases that operates with a shallow circuit, defined as a quantum circuit that does not scale with system sizes. Starting from a periodic circuit that simulates the dynamical evolution of a Floquet system, we introduce quasiperiodicity to the circuit parameters to prevent thermalization by introducing many-body localization. By inspecting the time-averaged properties of the many-body integrals of motion, the phase structure can then be probed using random measurements. This approach avoids the need to compute the ground state and operates at finite energy density. We numerically demonstrate this scheme with a simulation of the Floquet Ising model of time crystals and present results clearly distinguishing different Floquet phases that are protected by many-body localization. Our results pave the way for mapping out phase diagrams of exotic systems on near-term quantum devices.
Recent breakthroughs have opened the possibility of intermediate-scale quantum computing with tens to hundreds of qubits, and shown the potential for solving classical challenging problems, such as in chemistry and condensed matter physics. However, the high accuracy needed to surpass classical computers poses a critical demand on the circuit depth, which is severely limited by the non-negligible gate infidelity, currently around 0.1%-1%. The limited circuit depth places restrictions on the performance of variational quantum algorithms (VQA) and prevents VQAs from exploring desired nontrivial quantum states. To resolve this problem, we propose a paradigm of Schrödinger-Heisenberg variational quantum algorithms (SHVQA). Using SHVQA, the expectation values of operators on states that require very deep circuits to prepare can now be efficiently measured by rather shallow circuits. The idea is to incorporate a virtual Heisenberg circuit, which acts effectively on the measurement observables, into a real shallow Schrödinger circuit, which is implemented realistically on the quantum hardware. We choose a Clifford virtual circuit, whose effect on the Hamiltonian can be seen as efficient classical processing. Yet, it greatly enlarges the state's expressivity, realizing much larger unitary t designs. Our method enables accurate quantum simulation and computation that otherwise are only achievable with much deeper circuits or more accurate operations conventionally. This has been verified in our numerical experiments for a better approximation of random states, higher-fidelity solutions to the XXZ model, and the electronic structure Hamiltonians of small molecules. Thus, together with effective quantum error mitigation, our work paves the way for realizing accurate quantum computing algorithms with near-term quantum devices.
We investigate the use of variational quantum algorithms to prepare and characterize fractional quantum Hall states on near-term quantum processors. Focusing on the ν=1/3ν=1/3ν=1/3 Laughlin phase described by the V1V_1V1​ Haldane pseudopotential, we formulate the lowest-Landau-level problem in second quantization, and implement particle-number-preserving variational circuits combined with the variational quantum eigensolver (VQE) and variational quantum deflation (VQD). We benchmark the approach in two complementary geometries: Haldane sphere and torus shape. On the Haldane sphere, the target state is a unique zero-energy Laughlin ground state, providing a controlled test of the variational workflow and of excited-state reconstruction. On the torus, the problem retains the genuinely two-dimensional periodic character of the quantum Hall liquid and exhibits the threefold topological ground-state degeneracy expected for the ν=1/3ν=1/3ν=1/3 fractional filling factor. This feature makes the torus a more demanding benchmark than the quasi-one-dimensional cylinder or thin-torus limits commonly exploited in state-preparation quantum protocols. We benchmark the hardware-optimized variational states against exact diagonalization using energy estimates, error-mitigated observables, and subspace-containment diagnostics. Our results show that hybrid quantum algorithms can approximately reconstruct the low-energy structure of small fractional quantum Hall systems, including the topological ground-state manifold on the torus. Beyond serving as a benchmark for quantum hardware, this geometry-resolved approach provides a route toward quantum simulations of fractional Chern insulators and strongly correlated topological phases in realistic two-dimensional materials.
Solving non-Hermitian quantum many-body systems on a quantum computer by minimizing the variational energy is challenging as the energy can be complex. Here, we propose a variational quantum algorithm for solving the non-Hermitian Hamiltonian by minimizing a type of energy variance, where zero variance can naturally determine the eigenvalues and the associated left and right eigenstates. Moreover, the energy is set as a parameter in the cost function and can be tuned to scan the whole spectrum efficiently by using a two-step optimization scheme. Through numerical simulations, we demonstrate the algorithm for preparing the left and right eigenstates, verifying the biorthogonal relations, as well as evaluating the observables. We also investigate the impact of quantum noise on our algorithm and show that its performance can be largely improved using error mitigation techniques. Therefore, our work suggests an avenue for solving non-Hermitian quantum many-body systems with variational quantum algorithms on near-term noisy quantum computers.