Provably Robust Training of Quantum Circuit Classifiers Against Parameter Noise

TL;DR

The paper tackles robust quantum classification under parameter noise on NISQ devices by introducing a provably noise-resilient training framework for parameterized quantum circuit (PQC) classifiers. Central to the approach is a smoothed PQC classifier $G_{\sigma}$ and a Noise-resilient Theorem that provides a bound, $\|\delta \oslash \sigma\|_2 < \frac{1}{2} (\Phi^{-1}(p_A) - \Phi^{-1}(p_B))$, guaranteeing unchanged predictions under perturbations. The method combines Evolutionary Strategies-based optimization to maximize a robustness bound by jointly tuning $\theta$ and $\sigma$, and includes Variance Regularization to balance robustness with accuracy. Empirical evaluation on quantum phase classification with a QCNN demonstrates meaningful robustness certificates and reveals insights into per-parameter noise sensitivity, offering a practical pathway to robust near-term quantum computing. The work lays a foundation for certified robustness in PQCs and points to future extensions such as full-covariance smoothing and applications to other quantum algorithms like VQE or QAOA.

Abstract

Advancements in quantum computing have spurred significant interest in harnessing its potential for speedups over classical systems. However, noise remains a major obstacle to achieving reliable quantum algorithms. In this work, we present a provably noise-resilient training theory and algorithm to enhance the robustness of parameterized quantum circuit classifiers. Our method, with a natural connection to Evolutionary Strategies, guarantees resilience to parameter noise with minimal adjustments to commonly used optimization algorithms. Our approach is function-agnostic and adaptable to various quantum circuits, successfully demonstrated in quantum phase classification tasks. By developing provably guaranteed optimization theory with quantum circuits, our work opens new avenues for practical, robust applications of near-term quantum computers.

Figures (3)

• Figure 1: Accuracy over 20 test data points for a well-trained PQC and smoothed PQC with varying levels of noise added to the parameters. Each point and bar pair indicates the mean and confidence interval of 100 noisy-parameter samples from a gaussian with variances corresponding to $\sigma$ of the smoothed classifier multiplied by various scaling constants. "Noise Norm" is the average L2-norm of the noise sampled that produced each point.
• Figure 2: Phase classification for the generalized cluster Hamiltonian of 12 qubits, as outlined in Section \ref{['sec:phase_class']}. The first row illustrates the trade-off between accuracy and robustness, as described in Section \ref{['sec:robust_acc_front']}. The last row shows the robustness-variance correlation, as described in Section \ref{['sec:semi-axis-var']}. While our results may vary due to randomness and instability in optimization, we include a linear fit line to indicate the general trend.
• Figure 3: Figure 2 from gil-fuster_understanding_2023. Illustrates the different phases of the generalized cluster phase-classification problem outlined in Section \ref{['sec:phase_class']}.

Theorems & Definitions (6)

• Definition 2.1: PQC Classifier
• Definition 2.2: Noise-resilient PQC Classifier
• Definition 2.3: Smoothed PQC Classifier
• Theorem 2.1: Noise-resilient Condition
• Theorem C.1: Noise-resilient Condition for Smoothed PQC Classifier
• Lemma C.1