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Floquet-ADAPT-VQE Targets Quasienergies in Driven Systems
Floquet-ADAPT-VQE recasts periodically driven quantum systems as a static problem to calculate quasienergies and Floquet states. The paper reports classical simulations on driven XYZ spin chains and describes fixed-depth circuits for evalua
Quantum.world Team · Written with AI · fact-checked ·
Simulating a periodically driven quantum system usually means tracking its evolution through time, but existing quantum algorithms can bring extra circuit costs or work only in restricted drive regimes. A paper published in Quantum on October 23 introduces Floquet-ADAPT-VQE, a hybrid approach that recasts the repeating dynamics as a static problem and targets the system’s quasienergies and states. The authors test it in classical simulations of driven XYZ spin chains, not on a quantum processor.
Trading time dependence for an auxiliary register
Floquet theory describes periodically driven systems through quasienergies and Floquet states. A quasienergy plays a role like energy, but it is defined modulo the drive frequency; the periodic Floquet state captures the motion within each drive cycle. Those quantities can help characterize driven phenomena such as time crystals and Floquet topological phases.
The authors encode Fourier modes of the periodic drive in an auxiliary quantum register alongside the physical system. In that expanded Floquet-Hilbert space, they construct a time-independent Hamiltonian and decompose it into Pauli operators, the form needed for qubit-based algorithms. The trade-off is a finite truncation of the Fourier modes: the paper says lower drive frequencies and stronger drives generally require retaining more modes, and therefore more auxiliary qubits.
An adaptive search for the Floquet spectrum
Floquet-ADAPT-VQE combines the variational quantum eigensolver with ADAPT-VQE, which builds a circuit ansatz iteratively by selecting operators from a pool. Rather than minimizing the extended Hamiltonian’s energy directly, the method minimizes the square of a shifted Hamiltonian, with the shift parameter used to target quasienergies. Scanning that parameter lets the authors calculate the spectrum within a Floquet zone.
“A particularly important feature of our framework is that it enables access to micromotion dynamics of a Floquet state using fixed-depth quantum circuits.”
The authors—Abhishek Kumar, Karunya Shirali, Nicholas J. Mayhall, Sophia E. Economou and Edwin Barnes—write
The framework also uses a specially chosen starting state for the auxiliary register: an eigenstate of a diagonal auxiliary matrix with zero eigenvalue. The authors prove that, with this choice, the initial squared Floquet energy does not depend on the number of auxiliary qubits or the drive frequency. Their simulations show better convergence and fewer ADAPT iterations than with random starting states.
XYZ-chain tests show both promise and limits
The team’s numerical tests use periodically driven XYZ spin chains in a magnetic field. Across the reported cases, the calculated quasienergies closely match reference values, especially at lower drive amplitudes. At moderate frequencies, the relative error grows as the drive amplitude increases; the paper says adding auxiliary qubits can improve accuracy. Higher-frequency cases need fewer auxiliary qubits in the tested model.
The comparison is not a hardware demonstration. The authors run classical simulations, and their reference quasienergies come from diagonalizing the one-period evolution operator, which they compute using Trotterization. An additional test on a seven-qubit XYZ model finds one central-zone quasienergy for each tested magnetic-field drive amplitude, rather than establishing broad performance across larger systems.
Fixed circuit depth does not mean zero readout cost
Once an extended Floquet state has been prepared, the method can calculate time-dependent observables with a circuit whose depth stays fixed as the time within a drive period changes. The paper demonstrates calculations of net magnetization and a nearest-neighbor Z correlation. That removes the need to lengthen the circuit in proportion to the time being evaluated, a contrast with direct time evolution using Trotter steps.
Readout still has a cost: preparing the physical Floquet state at an arbitrary time requires measuring the auxiliary register and postselecting an outcome, with success probability 1/2 to the power of the number of auxiliary qubits. The authors’ resource analysis also estimates that, for a restricted operator pool and a fixed interaction-to-frequency ratio, gradient-measurement cost scales as O(log²(L) L⁶), where L is the physical-system size. The paper supplies an algorithmic route and classical tests; whether its circuit and measurement demands suit quantum hardware remains untested.