Quantum computing Floquet energy spectra

Benedikt Fauseweh1 and Jian-Xin Zhu2,3

1Institute for Software Technology, German Aerospace Center (DLR), Linder Höhe, 51147 Cologne, Germany
2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
3Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

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Quantum systems can be dynamically controlled using time-periodic external fields, leading to the concept of Floquet engineering, with promising technological applications. Computing Floquet energy spectra is harder than only computing ground state properties or single time-dependent trajectories, and scales exponentially with the Hilbert space dimension. Especially for strongly correlated systems in the low frequency limit, classical approaches based on truncation break down. Here, we present two quantum algorithms to determine effective Floquet modes and energy spectra. We combine the defining properties of Floquet modes in time and frequency domains with the expressiveness of parametrized quantum circuits to overcome the limitations of classical approaches. We benchmark our algorithms and provide an analysis of the key properties relevant for near-term quantum hardware.

In this study we introduce two quantum algorithms, Fauseweh-Zhu-1 and Fauseweh-Zhu-2, designed to compute Floquet eigenstates and quasi-energies. These algorithms employ parameterized quantum circuits in a quantum-classical hybrid methodology, aiming to variationally approximate Floquet eigenstates in both time and frequency domains. The precision of the first algorithm hinges on the depth of the quantum circuit, while the second algorithm's precision largely relies on frequency truncation and the width of the parameterized quantum circuit.

Both algorithms are equipped to operate on Noisy Intermediate-Scale Quantum (NISQ) devices, offering complementary circuit depth and width requirements as the system size increases. These were successfully tested on a quantum computer simulator, where we modeled a linearly driven spin-1/2 system, illustrating the practical feasibility of our approach. Further testing with a circularly driven spin-1/2 chain up to 8 sites demonstrated the scalability of our approach and shows a connection between variationally approximating ground state properties of critical systems and Floquet modes.

In addition, the effect of noise was simulated, which showed the resilience of our variational approach towards such perturbations. The parametrized quantum circuit's design is vital to the success of these hybrid algorithms. Interestingly, the second algorithm's qudit-qubit structure suggests a need for exploration of new schemes. Future work involves exploring more complex driving schemes and testing the performance on real devices with advanced error mitigation methods.

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Cited by

[1] Diogo Cruz and Duarte Magano, "Superresolution of Green's functions on noisy quantum computers", Physical Review A 108 1, 012618 (2023).

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