High-fidelity quantum gate learning via physics-informed neural networks

Introduction

The quest for ever‑more precise quantum control underpins the scalability of quantum computers. A recent study demonstrates that physics‑informed neural networks (PINNs) can be harnessed to design control fields that implement quantum gates with exceptionally low error rates. By training the network directly on the time‑dependent Schrödinger equation and incorporating physical constraints, the authors achieve average gate infidelities well below 10⁻⁶ for both single‑qubit (Hadamard) and two‑qubit (CNOT) operations.

Embedding Quantum Dynamics in a Neural Network

The core idea is to treat the control pulse shapes as trainable functions within a neural network. The loss function combines several components:

1. Model dynamics – the residual of the Schrödinger equation evaluated on a grid of time points.
2. Gate error – the deviation between the unitary produced by the learned control and the target gate, measured via average gate fidelity.
3. Physical constraints – smoothness, fixed boundary conditions, and limits on available local fields.
4. Regularization – penalties that discourage overly complex pulse shapes.
5. Multiple initial conditions – training on pairs of mutually unbiased bases to ensure the control works universally across the Hilbert space.

Training proceeds by minimizing this composite loss, yielding smooth control fields that respect the underlying physics while driving the system toward the desired unitary transformation.

Performance on Benchmark Gates

Using unitary simulations, the PINN‑derived controls achieve:

• Hadamard gate: average infidelity of \(8.49\times10^{-8}\).
• CNOT gate under three interaction types (XX, YY, ZZ): infidelities ranging from \(3.39\times10^{-7}\) to \(4.96\times10^{-6}\).

These numbers are obtained by validating the learned pulses on Haar‑random input states and computing the average gate fidelity, a standard metric for quantum gate quality.

Robustness to Dissipation

To assess practical viability, the authors test the learned controls against open‑system dynamics modeled by the Lindblad master equation with dephasing and amplitude‑damping channels. In the weak‑dissipation regime, the observed fidelity loss matches the theoretical first‑order prediction, indicating that the PINN‑generated pulses retain their high performance even when realistic noise is present.

Compact Representation of Controls

Beyond raw pulse shapes, the study explores efficient storage and playback of the learned controls. By fitting the fields with boundary‑adapted Chebyshev expansions, each control can be represented with as few as 24 coefficients while keeping the mean‑squared error below \(10^{-6}\). Additionally, square‑pulse and Strang‑splitting propagation schemes with modest resolution (e.g., \(K=30\) Chebyshev modes) reproduce the continuous‑time references within an infidelity of \(10^{-5}\) across a range of discretizations.

Practical Takeaways

• Physics‑aware training yields control fields that are both high‑fidelity and physically realizable, avoiding the need for post‑hoc filtering.
• Multi‑state training (using mutually unbiased bases) ensures that the learned pulses are not over‑fitted to a particular input state.
• Compact encoding via Chebyshev bases enables efficient deployment on hardware with limited memory or bandwidth.
• Noise resilience is built‑in, as the PINN framework naturally incorporates dissipative dynamics during validation.

Conclusion

Embedding the Schrödinger equation into a neural‑network training loop provides a powerful route to synthesize quantum gates with unprecedented precision. The approach bridges the gap between abstract optimal‑control theory and practical pulse engineering, offering a scalable pathway for high‑fidelity quantum operations in near‑term devices.