Walsh-Floquet Theory of Periodic Kick Drives

Description

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.

Technical info

# Walsh-Floquet Theory of Periodic Kick Drives

This Zenodo record contains the complete codebase, data, and reproducibility materials for the manuscript "Walsh-Floquet Theory of Periodic Kick Drives" by James Walkling and Marin Bukov. The repository provides all source code, analysis notebooks, generated data, and publication figures necessary to reproduce the results presented in the paper.

## Citation

### Plain Text Citation
J. Walkling and M. Bukov, “Walsh-Floquet Theory of Periodic Kick Drives,” arXiv:2505.11071 (2025).

### BibTeX Citation
```bibtex
@article{Walkling2025,
  author       = {James Walkling and Marin Bukov},
  title        = {Walsh-Floquet Theory of Periodic Kick Drives},
  journal      = {arXiv preprint},
  eprint       = {2505.11071},
  archivePrefix= {arXiv},
  primaryClass = {quant-ph},
  year         = {2025},
  month        = may,
}
```

## Links
- **arXiv**: https://doi.org/10.48550/arXiv.2505.11071
- **Zenodo DOI**: https://doi.org/10.5281/zenodo.17790671

## Table of Contents
- [Repository Overview](#repository-overview)
- [Installation and Setup](#installation-and-setup)
- [Directory Structure](#directory-structure)
- [Data Formats and Organization](#data-formats-and-organization)
- [Reproducing Results](#reproducing-results)
- [Figure Generation](#figure-generation)
- [License Information](#license-information)

## Repository Overview

This repository provides the associated code framework to implement the Floquet extended Hilbert space formalism in the Walsh basis.

## Installation and Setup

### Prerequisites
- Python 3.8 or higher
- Git for repository management
- At least 8GB RAM (16GB+ recommended for larger systems)

### Environment Setup
Create a dedicated Python environment and install all required packages.

## Directory Structure

```
WalshFloquet/
├── README.md                   # This file
├── requirements.txt            # Python package dependencies
├── environment.yml             # yaml file to create python environment
├── LICENSE                     # BSD 3-Clause license for code
├── LICENSE-DATA                # CC-BY 4.0 license for data
├── src_code/                   # Source code implementation
│   ├── README.md               # Source code documentation
│   ├── src/                    # walsh_floquet library
│   └── scripts/                # Standalone execution scripts
├── data/                       # Generated and processed data
│   ├── README.md              # Data organization documentation
│   └── processed/              # Processed data for plotting
├── visual_elements/            # Publication figures and assets
│   ├── README.md               # Figure documentation
│   ├── figs/                   # Manuscript figures
└── src_latex/                  # LaTeX source for manuscript
```

## Data Formats and Organization

### Data Formats
Data are given in **pkl** format for easy i/o in python.

## Reproducing Results
For the relative directory navigation to the walsh_floquet library to work, scripts need to be run from within their folder.

### Complete Reproduction Pipeline

1. **Environment Setup**: Follow installation instructions above
2. **Data Generation**: Run computational scripts to generate raw data. 
3. **Visualization**: Generate figures using plotting scripts

## Figure Generation

Figures can be regenerated using Python scripts or svg files. Further interactive exploration of results is also available.
Some figures were created using Inkscape, for which the **svg** file is provided.

### Figures
All figures together with scripts are placed at `visual_elements/figs/`.

- **Figure 1** `loc_schematic.svg` (SVG) -> `loc_schematic.pdf`
  **Schematic representation of localisation of the response to a periodic drive in terms of modes in the Fourier and Walsh bases**

- **Figure 2** `walsh_examples.svg` (SVG) -> `walsh_examples.pdf`
  **Example plots of the Walsh functions with corresponding Hadamard matrix and tip of a sawtooth wave expanded in Fourier vs. Walsh basis**

- **Figure 3** `spin_kick.py` (python) -> `spin_kick.pdf`
  **Single spin quasienergy spectrum and truncation error scan for kick drive in Walsh and Fourier bases**

- **Figure 4** `mb_spin_kick.py` (python) -> `mb_spin_kick.pdf`
  **Many spin quasienergy spectrum and truncation error scan for kick drive in Walsh and Fourier bases**

- **Figure 5** `localisation_vs_error.py` (python) -> `localisation_vs_error.pdf`
  **Localisation on frequency lattice along with comparing error vs. localisation for single kicked spin**

- **Figure 6** `alias_sqwave_res_polariton.py` (python) -> `alias_sqwave_res_polariton.pdf`
  **Schematic of aliasing, square wave drive error/localisation, resonance effects and Walsh polariton in time domain**

- **Figure 7** `discrete_generator.py` (python) -> `discrete_generator.pdf`
  **Generator of discrete cyclic translations in time domain (real space)**

- **Figure 8** `quasienergy_matrices.py` (python) -> `quasienergy_matrices.pdf`
  **Quasienergy matrices in Walsh and Fourier bases**

- **Figure 9** `error_scaling.py` (python) -> `error_scaling.pdf`
  **Scaling of errors with truncation for square wave and kick for single particle and many-body**

- **Figure 10** `error_scaling_strong_kick.py` (python) -> `error_scaling_strong_kick.pdf`
  **Scaling of errors with truncation for strong kick for single particle and many-body**
  
- **Figure 11** `kick_gauge.svg` (SVG) -> `kick_gauge.pdf`
  **Schematic of symmetric vs. non-symmetric kick drive gauge choice**

- **Figure 12** `walsh_series.py` (python) -> `walsh_series.pdf`
  **Scaling of the truncated series solution with omega**
  

## License Information

### Source Code
- **License**: [BSD 3-Clause License](./LICENSE)
- **Applies to**: All files in `src_code/` directory
- **Permissions**: Commercial use, modification, distribution
- **Requirements**: License and copyright notice

### Data
- **License**: [Creative Commons Attribution 4.0 International (CC-BY 4.0)](./LICENSE-DATA)
- **Applies to**: All files in `data/` directory
- **Permissions**: Share, adapt, commercial use
- **Requirements**: Attribution to original authors

---

**Repository Version**: v1.0
**Last Updated**: December 2nd 2025
**Zenodo DOI**: https://doi.org/10.5281/zenodo.17790671