Floquet-ADAPT-VQE: A Quantum Algorithm to Simulate Non-Equilibrium Physics in Periodically Driven Systems

Introduction

Periodically driven (Floquet) quantum systems host a wealth of phenomena—time crystals, Floquet topological phases, and exotic dynamical transitions—that have no equilibrium counterpart. Simulating their dynamics on classical computers quickly becomes intractable as system size grows, making them prime candidates for quantum advantage. The recent paper Floquet‑ADAPT‑VQE introduces a hybrid quantum‑classical framework that leverages the Floquet‑Hilbert space formalism to recast a time‑periodic Hamiltonian as a static operator in an enlarged space. By providing a systematic Pauli decomposition of this extended Hamiltonian and coupling it with an adaptive variational eigensolver, the authors obtain quasienergies, Floquet eigenstates, and time‑dependent observables with circuit depths that remain modest.

From Time‑Periodicity to a Time‑Independent Problem

The key insight is to embed the periodic drive into an auxiliary register that tracks the Fourier (Floquet) modes of the system. In this Floquet‑Hilbert space the original time‑dependent Schrödinger equation becomes an eigenvalue problem for a time‑independent Floquet Hamiltonian \(\mathcal{H}F\). The authors detail how to construct \(\mathcal{H}F\) from the physical Hamiltonian \(H(t)\) and derive its exact Pauli‑operator expansion, making it directly implementable on gate‑based quantum processors. Importantly, the auxiliary qubits do not increase the scaling of the required circuit depth because the Fourier mode structure is encoded analytically rather than via explicit time‑evolution.

Floquet‑ADAPT‑VQE: An Adaptive Variational Solver

Building on the static representation, the algorithm employs an adaptive variational quantum eigensolver (ADAPT‑VQE). Starting from a simple reference state, the method iteratively adds operators from a predefined pool that most steeply lower the energy functional—here the Floquet energy (the quasienergy). A notable contribution is the identification of an auxiliary initial state whose squared Floquet energy is independent of both the number of auxiliary qubits and the driving frequency. This choice improves convergence, often requiring fewer ADAPT iterations than a naïve product‑state initialization.

Once a Floquet eigenstate is prepared, the framework provides access to the full micromotion of the system. By applying a fixed‑depth circuit that implements the inverse Floquet transformation, one can evaluate expectation values of observables at arbitrary points within the driving period without increasing circuit depth with time. This contrasts with conventional Trotter‑based approaches where depth grows linearly with evolution time.

Demonstration on a Driven XYZ Spin Model

The authors validate their approach with numerical simulations of a periodically driven XYZ spin chain subjected to a transverse magnetic field. The Floquet‑ADAPT‑VQE algorithm accurately reproduces quasienergy spectra and captures characteristic Floquet signatures such as avoided crossings and band folding. Moreover, the computed time‑dependent magnetization profiles match exact diagonalization results, confirming that the micromotion extraction works as intended.

Outlook and Potential Applications

Floquet‑ADAPT‑VQE opens several avenues for future research. Because the method works for any Hamiltonian that can be expressed in the Floquet‑Hilbert space, it can be applied to study:

• Discrete time crystals and their stability against perturbations;
• Floquet topological phases, where edge‑state dynamics can be probed via the prepared Floquet eigenstates;
• Quantum thermalization and many‑body localization under periodic driving, by tracking quasienergy level statistics;
• Floquet engineering of synthetic gauge fields and interaction patterns in quantum simulators.

The algorithm’s reliance on shallow circuits makes it well‑suited for near‑term noisy intermediate‑scale quantum (NISQ) devices, while its adaptive nature mitigates barren‑plateau issues that often plague fixed‑ansatz variational methods.

Practical Takeaway

Floquet‑ADAPT‑VQE demonstrates that the powerful toolbox developed for static Hamiltonians—variational ansätze, adaptive operator selection, and efficient Pauli decomposition—can be transplanted to the realm of periodically driven systems. By converting time‑periodic problems into static ones in an extended Hilbert space, the method delivers quasienergy spectra, Floquet states, and observable dynamics with circuit depths that remain tractable on current quantum hardware. Researchers interested in non‑equilibrium quantum phases now have a concrete, hardware‑friendly pathway to explore these rich phenomena.