Designing Robust Quantum Gates

Yuanxing Hao and Marin Bukov Abstract For given control channels, quantum control theory determines whether quantum gates can be realized, but not their resilience to errors. We address this problem by establishing existence criteria for robust gate implementation in $d$-level systems. Our criteria reveal that implementability of all gates is not enough: a universal family of robust gates exists if and only if, in addition, the driven Hamiltonians share no common symmetry. When these criteria hold, we can make every gate resilient to any combination of static perturbations, uniform Hamiltonian rescaling, perturbations at a finite set of selected frequencies, and perturbations with arbitrary selected waveforms. We can impose robustness conditions broadly with limited information about errors, e.g., against any static perturbations; we can also tailor these conditions to the platform, e.g., against environment-induced noise with dominant spectral peaks, requiring no spectral details beyond the peak frequencies. We numerically construct explicit protocols for robust gates, reducing average gate infidelity by approximately two to seven orders of magnitude under Hamiltonian perturbations and environment-induced noise. Together, our existence criteria and numerical constructions provide practical guidance for evaluating the potential of quantum platforms and designing robust quantum gates. Technical info # Zenodo Record README ## Designing Robust Quantum Gates This repository contains the numerical code, data, optimized controls, paper's LaTeX source, and visual assets associated with Designing Robust Quantum Gates, a framework for establishing existence criteria and numerically constructing robust quantum gates. The study considers single-qubit, qutrit, and two-qubit systems. In these systems, we examine robustness against static perturbations, uniform time-stretching errors, perturbations at selected frequencies and with selected waveforms, thermal baths, and Lindblad noise. The archive helps readers identify the numerical datasets, optimized controls, calculation and plotting scripts, and paper files that support the main-text and appendix results. ## Table of Contents - Record Summary- Licensing- What This Record Contains- Data formats- Requirements and installation- Reproducing the figures ## Record Summary - `src_code/` contains control-optimization, quantum-dynamics, response-calculation, analysis, validation, and summary-generation scripts.- `data/` contains optimized controls, sampled Hamiltonians, response functions, gate-infidelity scans, inset data, and selected sixth-order bath-validation results.- `src_latex/` contains the paper's LaTeX source and compiled bibliography.- `visual_elements/` contains the paper figures and their plotting scripts.- `documents/` provides mathematical references and their source addresses, supplementary proofs, and diagnostic figures. In practical terms, the workflow is: ```control optimization + numerical analysis-> saved numerical results in data/processed-> plotting scripts and exported figures-> paper``` The numerical plotting scripts read the archived results directly without rerunning control optimization or quantum-dynamics calculations. ## Licensing The included license files specify the following scopes: - Original Python and shell source files in `src_code/scripts/` and `visual_elements/figs/`, including their subdirectories: BSD 3-Clause License; see `LICENSE`.- Project-generated numerical data in `data/`, PDF and PNG figures in `visual_elements/figs/` and `documents/figs/`, the numerical summary `src_code/config/control_resource_summary.txt`, the LaTeX source and bibliography in `src_latex/`, and project documentation: Creative Commons Attribution 4.0 International License (CC BY 4.0); see `LICENSE-DATA`. These license files specify licensing terms for the LaTeX source. The software dependencies in `requirements.txt` have their own upstream licenses. ## What This Record Contains ### Top-level directories | Path | Contents | Purpose || --- | --- | --- || `src_code/` | Numerical workflows, shared modules, and numerical summary | Control optimization, robustness analysis, validation, and summary generation || `data/` | Optimized controls, response functions, gate-infidelity scans, and validation results | Numerical inputs for analysis and plotting || `src_latex/` | Paper source and compiled bibliography | Paper preparation || `visual_elements/` | Paper figures and plotting scripts | Visual results from the numerical workflows || `documents/` | Mathematical references and source addresses, supplementary proofs, and diagnostic figures | Background references for the mathematical arguments | ### Main files and subdirectories ```.├── README.md├── requirements.txt├── vLOG.md├── LICENSE├── LICENSE-DATA├── src_code/│ ├── README.md│ ├── scripts/│ │ ├── README.md│ │ ├── bath_fourth_order.py # shared second- and fourth-order bath calculations│ │ ├── summarize_control_resources.py│ │ ├── validation/│ │ │ └── bath_sixth_order.py # shared sixth-order bath calculations│ │ ├── CZ-EGG/│ │ ├── CZ-DCG_ZZ/│ │ ├── CZ-DCG_all/│ │ ├── CZ-bath-DCG_ZZ/│ │ ├── CZ_Lindblad/│ │ └── appendix/ # appendix workflows│ ├── config/│ │ ├── README.md # numerical-summary guide│ │ └── control_resource_summary.txt│ └── log/ # cluster-log destinations; no run logs included├── data/│ ├── README.md│ ├── raw/ # optimization outputs│ └── processed/│ ├── CZ-EGG/│ ├── CZ-DCG_ZZ/│ ├── CZ-DCG_all/│ ├── CZ-bath-DCG_ZZ/│ ├── CZ_Lindblad/│ └── appendix/ # additional gate examples and bath spectra├── src_latex/│ ├── README.md│ ├── primary-article.tex│ └── primary-article.bbl├── visual_elements/│ ├── README.md│ └── figs/│ └── appendix/└── documents/ ├── REFERENCES.md # mathematical references and source addresses ├── README.md ├── independent_proofs_appendix_F.pdf └── figs/ # optimization and analysis diagnostic figures``` ## Data formats The numerical results and run metadata use the following formats: NumPy (`.npy`, `.npz`) — optimized controls, sampled Hamiltonians, response functions, gate-infidelity scans, inset data, and validation results. Read these files with `numpy.load`. Each `.npz` file contains multiple named arrays. Plain text (`.txt`) — numerical values, labeled coefficients, and summaries. JSON (`.json`) — optimization run settings. See `data/README.md` for the data organization. ## Requirements and installation ### Python Use Python 3.7–3.12. Install the dependencies from the project root: ```bashpip install -r requirements.txt``` For Python 3.7–3.9, key packages: `numpy` (1.21.6), `scipy` (1.7.3), and `matplotlib` (3.5.3); for Python 3.10–3.12: `numpy` (1.26.4), `scipy` (1.15.2), and `matplotlib` (3.10.9). ## Reproducing the figures The numerical plotting scripts read the supplied results from `data/processed/`. Run them from the project root, for example: ```bash# Main-text bath figurepython visual_elements/figs/figure_bath_DCG_ZZ.py # Appendix waveform-selective qutrit figurepython visual_elements/figs/appendix/figure_qutrit_general_functions.py``` The scripts write PDF and PNG files to `visual_elements/figs/` and its `appendix/` subdirectory. The proof-structure diagram uses `visual_elements/figs/appendix/figure_proof_structure.py` and requires no numerical data. See `visual_elements/README.md` for the complete figure list and `src_code/scripts/README.md` for calculation instructions.