Neural network approach to mitigating intra-gate crosstalk in superconducting CZ gates

1. Problem Statement

The practical realization of scalable quantum computing is fundamentally limited by crosstalk during multi-qubit gate operations. In superconducting transmon qubit systems, control pulses intended for one qubit often inadvertently affect neighboring qubits due to:

• Quantum Crosstalk: Intrinsic couplings (e.g., always-on $ZZ$ interactions) that shift energy levels and cause unintended entanglement.
• Classical Crosstalk: Unintended electromagnetic coupling where control signals leak into adjacent lines (cross-drive), causing spurious operations.

Existing mitigation strategies have significant limitations:

• Hardware isolation (spacing/shielding) increases fabrication complexity and can degrade individual qubit performance.
• Dynamical decoupling is sensitive to noise spectra and requires complex pulse sequences.
• Iterative calibration scales poorly with system size and is sensitive to hardware drift.
• Traditional Optimal Control (GRAPE, Krotov): While effective at a single nominal operating point, they struggle to find robust solutions under complex, varying crosstalk conditions, often getting stuck in local minima or failing to enforce hardware constraints (bandwidth, amplitude) simultaneously.

2. Methodology: Physics-Guided Neural Control (PGNC)

The authors propose a Physics-Guided Neural Control (PGNC) framework, a neural-network-based open-loop pulse generator designed to adapt to varying crosstalk conditions in real-time.

A. System Modeling

• Open-System Dynamics: The system is modeled as a Markovian open quantum system using the Lindblad master equation, accounting for $T_1$ (relaxation) and $T_2$ (dephasing) decoherence.
• Condition-Augmented Hamiltonian: The core innovation is modeling crosstalk as a function of a condition vector $\mathbf{c} = [c_I, c_Q, c_f]^\top$, representing concurrent-drive operating conditions (normalized neighbor-tone amplitudes and carrier offsets).
  • The Hamiltonian $H(t; \mathbf{c})$ includes condition-dependent terms: effective $ZZ$ coupling ($J_{zz}^{eff}$), cross-drive leakage coefficients ($r^{eff}$), and detuning biases ($\Delta \delta$).
  • These dependencies are modeled via a first-order affine map: $\mathbf{b}(\mathbf{c}) = \mathbf{b}_0 + G\mathbf{c}$, where $G$ is a sensitivity matrix.

B. Neural Network Architecture

• Input: Time $t$ (normalized to $[0,1]$) and the condition vector $\mathbf{c}$.
• Feature Embedding: Inputs are passed through a Fourier feature map to capture temporal modes and condition dependence.
• Network: A two-layer Multilayer Perceptron (MLP) with 64 hidden units (tanh activation) outputs a 7-dimensional latent vector.
• Output Mapping: The latent vector is mapped to physical control waveforms ($\Omega_{x,y}$, detuning $\delta$, and coupling $J_{zz}$) using:
  • Tanh Saturation: To enforce amplitude limits.
  • Analytic Flattop Envelope: To ensure smooth rise/fall times and zero endpoints, satisfying hardware bandwidth constraints structurally without post-processing.

C. Training Objective

The network is trained to minimize a composite loss function aggregated over a set of condition scenarios $\mathcal{C}$:
$$J_{tot} = \text{Agg} \left( \sum_{s} \left[ (1 - F(\mathbf{c}^{(s)})) + w_{leak} \cdot \text{Leak}(\mathbf{c}^{(s)}) + w_{smooth} \cdot \text{Smooth}(\mathbf{c}^{(s)}) \right] \right)$$

• Fidelity ($F$): Average gate fidelity over an ensemble of input states.
• Leakage: Penalty for population leaving the computational subspace.
• Smoothness: $H_1$-norm penalty on the time derivative of the waveform to reduce high-frequency jitter.
• Aggregation: The objective can use the mean (balanced) or CVaR (Conditional Value at Risk) to emphasize worst-case performance.

3. Key Contributions

1. Conditional Pulse Generation: Unlike traditional methods that output a single fixed waveform, PGNC learns a mapping $\mathbf{c} \to u(t; \mathbf{c})$. This allows a single trained network to generate optimal pulses for any crosstalk condition within the training distribution, eliminating the need for per-condition re-optimization.
2. Hardware-Aware Parameterization: The architecture inherently enforces physical constraints (amplitude limits, smooth edges, zero endpoints) through the output layer design, ensuring the generated pulses are experimentally realizable.
3. Multi-Channel Coordination: The framework discovers cooperative control strategies where transverse drives, detuning, and tunable coupling are modulated simultaneously to counteract crosstalk, rather than relying on a single dominant control knob.
4. Robustness via Implicit Regularization: Training across a distribution of conditions acts as an implicit regularizer, steering the solution toward "flat" regions of the control landscape that are less sensitive to parameter drifts.

4. Results

The authors benchmarked PGNC against Krotov and GRAPE (Gradient Ascent Pulse Engineering) baselines for a tunable-coupler CZ gate ($T=50$ ns) on a two-transmon system.

• Fidelity and Convergence:
  • PGNC and GRAPE converged rapidly to infidelities of $\sim 10^{-3}$, while Krotov converged slower with slightly higher residual infidelity.
  • PGNC achieved superior worst-case fidelity compared to baselines, particularly under perturbed conditions.
• Condition Generalization:
  • Static vs. Adaptive: When tested on off-nominal conditions (unseen during training), static baselines (Krotov/GRAPE) showed significant fidelity degradation and widened distributions. PGNC maintained near-unity fidelity by dynamically reshaping the pulse.
  • Robustness Sweeps: PGNC demonstrated a larger "safe zone" against static detuning drifts ($\delta_1, \delta_2$) compared to baselines.
• Waveform Characteristics:
  • PGNC utilized a multi-channel strategy, actively modulating the $y$-quadrature of the second qubit and coordinating it with detuning and coupling.
  • Krotov relied on large detuning plateaus and smooth $ZZ$ modulation.
  • GRAPE tended toward an "interaction-only" solution (dominant $J_{zz}$, near-zero transverse drives), which proved less robust to crosstalk-induced mixing.
• Leakage and Smoothness: PGNC produced smoother waveforms with lower leakage, attributed to the smoothness penalty and the structural enforcement of bandwidth limits.

5. Significance and Future Outlook

• Scalability: PGNC offers a path to scalable quantum control by amortizing the optimization cost. Once trained on an effective local model, the network can generate pulses for new conditions via a single forward pass, avoiding the computational bottleneck of iterative re-optimization for every gate configuration.
• Practicality: The approach is compatible with current calibration workflows. The condition vector $\mathbf{c}$ can be updated based on real-time calibration data (e.g., measured crosstalk levels), allowing the controller to adapt to hardware drift without retraining.
• Extension: The framework is extensible to inter-gate crosstalk by expanding the condition vector to include timing and spectral descriptors of neighboring gates, enabling robust control in dense, concurrent operation schedules.

Conclusion: The paper demonstrates that a physics-guided neural network can effectively mitigate intra-gate crosstalk in superconducting qubits, offering a robust, scalable, and hardware-compliant alternative to traditional optimal control methods, particularly in the presence of fluctuating noise and crosstalk conditions.