Geometric Optimization on Lie Groups: A Lie-Theoretic Explanation of Barren Plateau Mitigation for Variational Quantum Algorithms

TL;DR

This work tackles barren plateaus in variational quantum algorithms by introducing a Lie-theoretic viewpoint that maps single-qubit gates to the SU(2) group and interprets parameter updates as motions in the associated su(2) algebra. By employing neural-network-generated parameters, the evolution of gate parameters follows geodesic-like trajectories on SU(2), characterized by near-zero velocity and acceleration, which reduces gradient vanishing and accelerates convergence. The authors quantify this behavior using velocity v, acceleration a, trajectory energy E, and length L, demonstrating that neural-network-based updates yield smoother, more geodesic paths than non-NN updates across systems with 4–9 qubits. This geometric explanation informs how to design trainable quantum circuits and initialization strategies, and points to extending the approach to higher-dimensional representations and multi-qubit gates in future work.

Abstract

Barren plateaus, which means the training gradients become extremely small, pose a major challenge in optimizing parameterized quantum circuits, often making the learning process impractically slow or stall. This work shows why using neural networks to generate quantum circuit parameters helps overcome this difficulty. We introduce a geometric viewpoint that describes how the parameters produced by neural networks evolve during training. Our analysis shows that these parameters follow smooth and efficient paths that avoid the flat regions in the training that cause barren plateaus. This provides a computational explanation for the improved trainability observed in recent neural network-assisted quantum learning methods. Overall, our findings bridge ideas from quantum machine learning and computational optimization, offering new insight into the structure of quantum models and guiding future approaches for designing more trainable quantum circuits or parameter initialization.

Figures (3)

• Figure 1: Geometric representation of single-qubit rotation gates on the Bloch sphere. These trajectory represents the operation of $R_{x}(\theta),R_{y}(\theta),R_{z}(\theta)$ as the element of $\mathbb{SU}(2)$. The $R_{x}$ and $R_{y}$ is act on the $\ket{0}$ whill the $R_{z}$ is act on the $\ket{+}=\frac{\ket{0}+\ket{1}}{\sqrt{2}}$.
• Figure 2: Fluctuation comparison of the group-mapped value with and without the neural-network module. The x-axis represents the iteration number $k$, and the y-axis represents the group-mapped value $y$. We focus on the fluctuation pattern of $y$ across $k$. Using neural networks helps mitigate barren plateaus 1516, resulting in fewer iterations to convergence and a shorter trajectory.
• Figure 3: Comparison of the Velocity and Acceleration in Generating Parameters with Different Numbers of Qubits. The neural network-enhanced model (left/blue usually) shows velocity and acceleration closer to zero, indicating geodesic-like stability.

Theorems & Definitions (1)

• proof