Walsh-Floquet Theory of Periodic Kick Drives

TL;DR

This paper introduces Walsh-Floquet theory, demonstrating that using Walsh basis functions improves the accuracy of modeling periodic kick drives in quantum systems, especially in strongly kicked regimes, compared to traditional Fourier methods.

Contribution

The paper develops Walsh-Floquet theory, including an extended Sambe space and inverse-frequency expansion, providing a more accurate framework for analyzing periodic quantum drives.

Findings

Walsh basis improves convergence over Fourier basis in strongly kicked regimes.

Walsh polaritons emerge from hybridization with Walsh modes, enabling quantum simulation.

Localization on the frequency lattice explains truncation error behavior.

Abstract

Periodic kick drives are ubiquitous in digital quantum control, computation, and simulation, and are instrumental in studies of chaos and thermalization for their efficient representation through discrete gates. However, in the commonly used Fourier basis, kick drives lead to poor convergence of physical quantities. Instead, here we use the Walsh basis of periodic square-wave functions to describe the physics of periodic kick drives. In the strongly kicked regime, we find that it recovers Floquet dynamics of single- and many-body systems more accurately than the Fourier basis, due to the shape of the system's response in time. To understand this behavior, we derive an extended Sambe space formulation and an inverse-frequency expansion in the Walsh basis. We explain the enhanced performance within the framework of single-particle localization on the frequency lattice, where localization is correlated with small truncation errors. We show that strong hybridization between states of the kicked system and Walsh modes gives rise to Walsh polaritons that can be studied on digital quantum simulators. Our work lays the foundations of Walsh-Floquet theory, which is naturally implementable on digital quantum devices and suited to Floquet state manipulation using discrete gates.