The PID Controller Strikes Back: Classical Controller Helps Mitigate Barren Plateaus in Noisy Variational Quantum Circuits

TL;DR

This work addresses the barren plateau problem in variational quantum algorithms by introducing a neural PID (NPID) controller that dynamically updates variational circuit parameters. A neural network adapts PID gains to current, integral, and derivative error signals, and the resulting control signal modulates gradient-based updates, enabling faster and more stable convergence. Across random input states, random noisy circuits, and varying qubit counts (7–12) and depths, NPID outperforms neural-enhanced quantum parametric models and standard quantum vanilla approaches, with 2–9x improvements in convergence efficiency and robust performance under parameter noise (average fluctuations ~4.45%). This work demonstrates the potential of integrating classical control theory with quantum optimization to improve the trainability of quantum neural networks in the NISQ era.

Abstract

Variational quantum algorithms (VQAs) combine the advantages of classical optimization and quantum computation, making them one of the most promising approaches in the Noisy Intermediate-Scale Quantum (NISQ) era. However, when optimized using gradient descent, VQAs often suffer from the vanishing gradient problem, commonly known as the barren plateau. Various methods have been proposed to mitigate this issue. In this work, we propose a hybrid approach that integrates a classical proportional-integral-derivative (PID) controller with a neural network to update the parameters of variational quantum circuits. We refer to this method as NPID, which aims to mitigate the barren plateau. The proposed algorithm is tested on randomly generated quantum input states and random quantum circuits with parametric noise to evaluate its universality, and additional simulations are conducted under different noise rates to examine its robustness. The effectiveness of the proposed method is evaluated based on its convergence speed toward the target cost value. Simulation results show that NPID achieves a convergence efficiency 2-9 times higher than NEQP and QV, with performance fluctuations averaging only 4.45% across different noise levels. These results highlight the potential of integrating classical control theory into quantum optimization, providing a new perspective for improving the trainability and stability of variational quantum algorithms.

Figures (6)

• Figure 1: The construction of the Variational Quantum Algorithm.
• Figure 2: The Construction of the random quantum circuit. The gray boxes represent individual circuit layers, while the black dashed boxes indicate the gates applied between pairs of randomly selected qubits.
• Figure 3: The construction of the Neural Proportional–Integral–Derivative Model.
• Figure 4: Average loss versus number of iterations. The shaded regions represent the mean $\pm$ standard deviation over five independent runs.
• Figure 5: Number of iterations needed for the target cost function for different numbers of qubits.