Walsh-Floquet Theory of Periodic Kick Drives

TL;DR

This work tackles the poor convergence of Floquet analyses based on Fourier basis for strongly kicked periodic drives by introducing the Walsh basis of periodic square-wave functions. It develops an extended Sambe-space formalism with a discrete time-translation generator to diagonalize the quasienergy operator in the Walsh basis, and introduces an inverse-frequency expansion tailored to this basis. The authors show that Walsh representations localize the Floquet modes on the frequency lattice more efficiently than Fourier under strong kicking, leading to markedly smaller quasienergy errors in single-particle and many-body Ising-type models and revealing Walsh polariton phenomena. The framework promises practical advantages for digital quantum simulators and gate-based Floquet engineering, while also highlighting regimes where Fourier-machine intuition remains advantageous, such as square-drive scenarios.

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.

Figures (12)

• Figure 1: (a) A kicked system (represented as a spin-$1/2$) obeying the Schrödinger equation generically has a square wave response in the observables, via the micromotion $\ket{u_n(t)}$. (b) frequency lattice states in the tensor product of the physical spin Hilbert space, $\mathcal{H}$, and the space of periodic functions, $\mathcal{L}_\odot$ [schematic]. Depending on the choice of basis, the response is localized on the frequency lattice labeled by mode number, $m$. Wavefunctions are defined by $\braket{\uparrow |u_n(t)} = \sum \psi_f(m) f_m(t)$ where $f_m(t)$ represents Fourier or Walsh modes.
• Figure 2: (a) Walsh basis functions for $N=4$ basis elements along with related trigonometric functions in a dashed line. This correspondence only works for $N=4$ since the roots of the Walsh functions are not evenly spaced for $N > 4$. (b) construction of the Walsh basis from the Hadamard matrix for $N=4$. (c) example of representing a function with discontinuities in different bases. The Walsh (green) consistently undershoots for smooth variation; by contrast, the Fourier (orange) oscillates wildly near sharp discontinuities (Gibbs phenomenon) for a periodic sawtooth wave (dashed black).
• Figure 3: Single-particle quasienergies from a kick drive, see Fig. \ref{['fig:IntroFig']}(a). (a) is the quasienergy spectrum for $\omega =10$ and $h_z=4.5$ showing the superior performance of the Walsh basis over the Fourier. (b) quantifies the error over a range of parameters, showing that Walsh outperforms Fourier the most for strong kick drives. In the orange region, the Fourier basis better approximates the quasienergies, and in the green region the Walsh performs better. $N=32(31)$ modes used in both plots.
• Figure 4: (a) Many-body quasienergy spectrum for $L=3$ spins and $N{=}32(31)$ modes with $J=1$ and $h_zT=1.1\pi$. The Walsh basis outperforms the Fourier. (b) Comparison of the error in the dimensionless quasienergy phases calculated using the Walsh and Fourier basis to the same level of truncation, $N{=}64(63)$ modes, for $L=6$ spins. Green indicates that Walsh is more accurate than the Fourier, and orange means the opposite. Apart from very small kick fields, the Walsh basis outperforms over a large parameter regime. At large kick fields, in the black region, there is an error w.r.t. the exact solution by more than 1% in both bases.
• Figure 5: (a) Spin-up component of the response of a kicked two-level system system, defined via $\braket{\uparrow| u(t)} = \sum_m \tilde{u}_m f_m(t)$, where $f_m(t)$ denotes the orthonormal basis (Fourier, Walsh), see legend. Main plot shows the modes; the inset -- their time evolution, which is close to a square wave at high frequencies. Dashed (solid) lines indicate the low- (high-) frequency regime. At high frequency, the signal is strongly localized in the Walsh basis, and follows a $1/m$ power law decay otherwise. (b) difference in quasienergy errors, $\langle \Delta \theta \rangle \equiv\Delta \theta_{FW}=|\Delta \theta_F|- |\Delta \theta_W|$, and photon participation entropy ($\Delta S$ Eq. \ref{['eqn:PPE']}) highlighting the similarity between localization and associated error, scaled by a factor of $10^2$ (see text box). The errors are computed over all spins; localization is only shown for the spin-up component. The singular behavior near $h_zT=0$ comes from the degeneracy of $U(T,h_z=0)=\mathbf{1}.$ Data obtained using Eq. \ref{['eqn:MFIM']} with $L=1$; for (a) $\omega, h_x= 10, \pi/2$.