The model captures parameter dependence while preserving unitary structure and maintaining accuracy over long-time coherent evolution. Quantum Evolution Networks (QENs) map physical parameters and time to complete propagators for periodically driven three-spin registers. Shared networks learn unitary one-period propagators and compatible within-period paths, enabling predictions through known periodic composition without query-specific numerical integration. For transverse driving, the fifth-percentile fidelity is 0.9947 over 24 periods on 256 held-out configurations. A leading-order analysis of spectrally resolved endpoint data provides further insight into network behaviour.

## Neural Residuals Capture Quantum Dynamics Reliability and Predict Multicycle Errors Residual errors predict multicycle process error with greater precision than quadratic or spectrum-averaged estimates. Interventions targeting matched single-period infidelity produce orders-of-magnitude differences in multicycle infidelity, stressing the importance of residual orientation. Compatible component replacements identify how endpoint and within-period approximations limit reconstruction accuracy.

These results establish a reusable neural representation for studied propagator families, connecting its learned residual structure to observed long-time reliability. Predicting and controlling driven quantum systems requires repeated evaluation of dynamics across physical parameters, initial states, and evolution times; therefore parameter exploration and control design benefit from models that can answer many related dynamical queries. Machine learning offers an avenue for encoding this dependence into shared representations, for tasks requiring coherent evolution of arbitrary input states, the complete propagator serves as a common object for these predictions.

Learning parameter dependence transforms a family of dynamical solutions into a reusable model. Neural representations have been developed for several aspects of quantum dynamics: neural quantum states encode many-body wavefunctions, while dynamical extensions describe evolving states and learned updates advance wavepackets. Propagator learning represents the transformation acting on initial states with applications to open-system dynamics and time-dependent unitary evolution.

Function-space approaches, including Fourier neural operators and DeepONet, provide a broader framework for learning solution maps. Related quantum applications learn effective Floquet generators, observable dynamics, operator growth, as well as propagators across driving protocols; these methods support different forms of reuse depending on their inputs, learning targets, and imposed physical structure. A complete propagator must consider both physical structure and prediction accuracy to serve as a reusable dynamical model.

In periodically driven systems, approximation errors in a learned one-period propagator can accumulate or partially cancel under repeated composition. Magnitude alone does not determine multicycle accuracy while agreement for selected initial states may leave more sensitive states unresolved. This raises the question: how do we construct a shared representation of complete propagators and connect its learned errors to long-time prediction accuracy.

We introduce Quantum Evolution Networks (QENs) for prescribed families of periodically driven Hamiltonians; within each family, a finite parameter vector λ specifies the physical configuration, with the overall query map being (λ, t) → Uλ(t). Shared neural weights predict a unitary one-period propagator alongside a compatible within-period path.

Known periodic composition then reconstructs evolution at queried times, including motion within each period. The resulting operator acts on arbitrary initial states without state-specific fitting, numerical propagation supplies training labels and evaluation references while trained model queries use neural evaluation, unitary decoding, and matrix composition without query-specific integration. We investigate periodically driven three-spin central spin registers comprising an electron coupled to two nuclei through exchange and longitudinal interactions; longitudinal driving preserves excitation sectors whereas modulated transverse driving mixes them providing distinct settings for constructing and testing the representation.

The parameter scales are motivated by silicon electron/nuclear registers where long coherence and high-fidelity control have been demonstrated. A tractable Hilbert space permits complete numerical references and direct minimisation of prediction fidelity over all input states allowing examination of both the learned operator and residual structure underlying its long-time accuracy.

To explain observed multicycle errors we resolve actual learned endpoint residuals in the reference Floquet eigenbasis, building on coherent error analysis yielding a leading order estimate of repeated endpoint process error. We further examine training choices demonstrating reuse of the learned representation in numerically verified control design connecting construction of a reusable complete propagator to physical explanation of its multicycle accuracy and limits.

QUANTUM EVOLUTION NETWORKS FOR PROPAGATOR FAMILIES

A shared map from physical parameters to evolution is defined: let λ ∈Λ specify a member of a fixed Hamiltonian family H(t; λ). A QEN approximates this map with shared neural weights, taking physical parameters and time as inputs returning a complete propagator which acts on any pure or mixed initial state without state-specific fitting.

Evaluations below assess its accuracy across the prescribed Hamiltonian family using the same trained network. The exchange and longitudinal terms share the same gm; longitudinal driving conserves excitation number permitting 19 block coordinates whereas transverse driving requires full-space coordinates utilising ten physical and waveform inputs.

Hamiltonians, analytic frames, and decoder conventions are specified in Appendix A defining two separately trained families at the same three-spin system size. distinguishes trained modules from prescribed periodic composition, analytic frame transformations enforcing unitarity stated endpoint conditions assessing dynamics through evaluations below.

The within period path connects identity to the endpoint while closed Floquet micromotion loop returns to it; predicting bU(t)|ψ0⟩for pure input or bU(t)ρ0 bU(t)† for mixed input allows assessment of error over all initial states via relative operator defined in Sec III separating two contributions to prediction error, spectral norm satisfies ∥bV c UF n −V U n F∥2 ≤∥bV −V ∥2 + n∥c UF −UF∥. This inequality follows a telescoping expansion of endpoint powers with the path entering once and endpoint discrepancy successive periods.

NUMERICAL SCOPE AND EMPIRICAL RELIABILITY CRITERIA

A. The longitudinal endpoint uses 65,536 training parameters while the path utilizes 32,768, both having validation members test members. Transverse component employs member training pool member validation pool. Ilias Magradze at City University of Hong Kong and collaborating institutions have demonstrated a major advance in modelling complex quantum behaviour by accurately predicting three spin register dynamics through compositional application of learned networks.

Analysis reveals residual errors within these networks are key indicators long term prediction reliability offering insights maintaining coherence intricate systems. Critically, targeted interventions addressing single period inaccuracies resulted in orders magnitude improvements to long term infidelity highlighting the importance of how errors are oriented within network’s predictions.

Swapping components and observing accuracy changes allowed researchers identify specific limitations affecting reconstruction precision demonstrating potential areas refinement. While this work demonstrates impressive accuracy predicting three spin register dynamics it represents a vital step towards modelling more complex systems but highlights ongoing tension between machine learning approaches to quantum control: the trade off between fidelity scalability. The team acknowledges limitations extending their method beyond current system sizes different types periodic driving broader applicability remains elusive without addressing how networks will perform larger Hilbert spaces where computational demands rapidly increase, despite acknowledged limits scaling up technique ever present challenge.

Quantum Evolution Networks (QENs) accurately predicted the behavior of periodically driven three-spin registers by learning and composing unitary propagators. The research demonstrates that analyzing residual errors within these learned models can more precisely predict long-term reliability than existing methods. Furthermore, correcting inaccuracies in single time periods substantially improved overall prediction accuracy, revealing how error orientation impacts performance.

👉 More information
🗞 Quantum Evolution Networks Learn Long-Time Coherent Dynamics in Driven Spin Registers
✍️ Puran Li and Condon Lau (City University of Hong Kong); Xiu-Hao Deng (Affiliation: Shenzhen International Quantum Academy)
🧠 ArXiv: https://arxiv.org/abs/2610.32156

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