🌀 Swappy: Anomalous Transport in U(1)-Symmetric Quantum Circuits

Reproducible reference implementation and dataset for the prethermal swappy regime in disordered U(1)-symmetric Floquet quantum circuits. This repository contains the notebooks, data, and reproduction code accompanying:

Alessandro Summer, Alexander Nico-Katz, Shane Dooley, and John Goold, “Anomalous transport in U(1)-symmetric quantum circuits,” npj Quantum Information 12, 32 (2026). doi:10.1038/s41534-025-01178-8

TL;DR. We construct a strongly disordered, U(1)-symmetric Floquet circuit from physically interpretable XXZ-like two-qubit gates. Its central feature is a genuinely discrete-time swappy regime near the generalized-SWAP line: an imperfect SWAP transfers most of a local magnetization excitation to the neighboring qubit while leaving a small remnant behind. We study it by time-evolving a single-site excitation. The resulting packet moves fast, almost ballistically (close to the SWAP point) and at the same time locally diffuses quickly. The overall result is that it spreads faster than in the ordinary ergodic regime. This phenomenon has no direct continuous-time XXZ counterpart. To follow the packet across periodic boundaries, we introduce a complex circular moment whose phase measures directed motion and whose magnitude measures spreading using only local Z-basis expectation values.

Keywords: Swappy · swappy regime · quantum transport · Floquet circuits · U(1) symmetry · disorder · statistics · localization · digital quantum simulation.

Start here: Conceptual guide for researchers and LLMs · Reproducibility guide

🏖️ The picture in one glance

🎛️ Explore frame by frame: Open the slider and use ←/→ to step through the animation. Local alternative: run python -m http.server from swappy/, then open http://localhost:8000/docs/slider.html.

Planting a beach umbrella in disordered sand takes more than a downward push: a gentle twist helps the pole find a path, while excessive precession makes it spin without sinking. The animation uses this analogy to interpolate the discrete circuit samples $R_n$, mapping $\arg R_n$ to the umbrella’s orientation and $|R_n|$ to its height. Localized motion barely wanders, ergodic motion contracts almost radially, swappy motion spirals inward, and near-SWAP motion follows an orbit near the unit circle. The blue-to-red gradient marks time only, while inward contraction measures spreading—diffusive when $\sigma(t)\sim t^{1/2}$.

🎯 Main points

1. 🧭 A physically interpretable model is still a digital one.

   Every U(1)-symmetric gate has an XXZ-type exchange interaction $J$, an Ising interaction $J_z$, and independent random phases. The disorder stays order-one in the process of varying $J$ and $J_z$, thus small values of couplings do not quietly make the model a weak Trotter step for the continuous time. The same model can therefore be used to study the phenomena of localized-like, ergodic, swappy, and generalized-SWAP transport. Walkthrough

2. 🎯 Periodic evolution needs circular statistics.

   On a ring, the sites $N-1$ and $0$ are neighboring each other. However, an ordinary linear mean or variance would misinterpret a narrow packet that crosses this junction as being distributed on the entire lattice. Transformation of the magnetization distribution to quasi-probability $p_n(t)$ and further to $R(t)=\sum_n p_n(t)\exp 2\pi i n/N$ makes it possible to eliminate the effect of the boundary: the phase angle of $R$ follows drift on the ring, while contraction of $|R|$ follows spreading. The diagnosis requires only local Z-basis measurements. Analysis walkthrough

3. 🌀 The key observation is super-ergodic diffusion.

   Close to the generalized-SWAP line, each gate transfers most of the magnetization to the adjacent spin but retains a small portion. Thus, a coherent peak propagates ballistically, gradually leaving some of the excitation behind leading to a diffusion that is globally faster than ergodic one even in the presence of disorder. Entanglement and level statistics do not provide any information about the transient swappy regime: the moving coherent peak is present only in the dynamical properties. Dynamical reproduction

4. ⏳ Swappy transport is prethermal and purely discrete-time.

   The phenomenon utilizes the finite duration of near-$\pi$ exchange pulse: the almost-SWAP gate is responsible for the transport and split of local magnetization. It vanishes in the usual continuous-time limit and does not have any analog in the disordered continuous-time XXZ chain. Thermalization time increases proportionally to $t_{\mathrm{th}}\sim(\pi-J)^{-2}$ when approaching SWAP point. Reproducibility map

5. 🔬 Generalized SWAPs defeat disorder but are sensitive to detuning.

   At the generalized-SWAP line, random phases do not stop ballistic transfer: the excitation keeps circulating without thermalizing. A small detuning from perfect SWAP makes every hand-off slightly incomplete, so remnants accumulate and ultimately destroy the coherent peak. The investigated swappy window is therefore long-lived and robust over finite parameter and system-size ranges, but not an asymptotically stable phase.

🌀 How to read the circular moment

Clockwise from the upper left: localized (●), ergodic (▶), near-SWAP (★), and swappy (■).

Rotation of $R(t)$ tracks coherent drift of the excitation, while contraction toward the centre tracks its spreading. The ergodic trajectory therefore moves almost radially inward, the swappy trajectory spirals inward, and the near-SWAP trajectory remains close to the outer rim.

📓 The notebooks

1. 01_from_gate_to_swappy.ipynb Construct the U(1)-symmetric gate and watch the four regimes emerge.

2. 02_circular_moment_and_reproduction.ipynb Build the circular moment and reproduce the transport analysis.

Execute both:

make setup
make notebooks

▶️ Reproduce the results

For a laptop-scale reproduction:

make setup
make quick

This generates:

Paper result	Output
Static localization and ergodicity diagnostics	static_diagnostics.png
Four dynamical regimes	magnetization_regimes.png
Circular-moment trajectories	circular_moment.png
Spread, drift, peak decay, and coupling scan	transport_observables.png, phase_scan.png

The figures are written to artifacts/quick/. Verify the implementation with:

make test

For the published main dynamical settings:

make paper

The paper profile uses N=20, 100 disorder realizations, 10,000 Floquet cycles, and 33 values of $J$. It is an HPC-scale, resumable calculation. See docs/REPRODUCIBILITY.md for the exact figure map, conventions, and computational scope.

📚 Citation

Alessandro Summer, Alexander Nico-Katz, Shane Dooley, and John Goold, “Anomalous transport in U(1)-symmetric quantum circuits,” npj Quantum Information 12, 32 (2026). doi:10.1038/s41534-025-01178-8

Machine-readable citation metadata is available in CITATION.cff.

📁 Repository contents

notebooks/   guided explanation
src/swappy/  supported simulation and analysis
tests/       physics and reproducibility checks
docs/        detailed methods and figure map
data/        two archived static-data tables