[论文解读] Optimal training of variational quantum algorithms without barren plateaus

Tobias Haug, M. S. Kim|arXiv (Cornell University)|Apr 29, 2021

Quantum Computing Algorithms and Architecture参考文献 52被引用 25

一句话总结

本文引入用于变分量子算法的自适应梯度上升，利用高斯核 Fidelity–QFIM 关系，提出广义量子自然梯度以避免无梯度平原并加速训练，应用于量子控制和量子机器学习。

ABSTRACT

Variational quantum algorithms (VQAs) promise efficient use of near-term quantum computers. However, training VQAs often requires an extensive amount of time and suffers from the barren plateau problem where the magnitude of the gradients vanishes with increasing number of qubits. Here, we show how to optimally train VQAs for learning quantum states. Parameterized quantum circuits can form Gaussian kernels, which we use to derive adaptive learning rates for gradient ascent. We introduce the generalized quantum natural gradient that features stability and optimized movement in parameter space. Both methods together outperform other optimization routines in training VQAs. Our methods also excel at numerically optimizing driving protocols for quantum control problems. The gradients of the VQA do not vanish when the fidelity between the initial state and the state to be learned is bounded from below. We identify a VQA for quantum simulation with such a constraint that thus can be trained free of barren plateaus. Finally, we propose the application of Gaussian kernels for quantum machine learning.

研究动机与目标

• Motivate and address the slow training and barren plateau problems in variational quantum algorithms (VQAs).
• Develop an adaptive learning-rate scheme based on a Gaussian kernel of fidelity and the quantum Fisher information metric (QFIM).
• Introduce the generalized quantum natural gradient (GQNG) and identify a stable regime that avoids regularization.
• Demonstrate improved training efficiency for VQAs and numerical quantum control tasks.
• Discuss implications for quantum machine learning and practical instantiations on near-term hardware.

提出的方法

• Model the fidelity between PQC states as a Gaussian kernel with the QFIM as the weight matrix (Eq. 4).
• Define the generalized quantum natural gradient Gβ(θ)=F(θ)^(−β) ∇Kt(θ) with β∈[0,1] (Eq. 5).
• Derive adaptive learning rates per iteration based on the kernel: α1 and αt (Eqs. 7–8).
• Show stability conditions: Gβ is stable without regularization for β≤1/2, with regularization needed for β>1/2 (Eq. 6 discussion).
• Provide analytic expressions for gradient variance under fidelity bounds (Eq. 9–10).
• Apply methods to training PQCs for state learning and to driving protocols in quantum control (Eq. 12–14).

实验结果

研究问题

• RQ1 Can adaptive gradient ascent using a Gaussian fidelity kernel and the quantum Fisher information metric avoid barren plateaus in VQAs?
• RQ2 What is the optimal interpolation (β) between standard gradient and quantum natural gradient for stable and efficient training?
• RQ3 How do adaptive learning rates affect convergence speed and fidelity in VQAs and quantum control problems?
• RQ4 Can the proposed methods systematically improve quantum control protocol optimization compared to standard solvers?

主要发现

• The fidelity between PQC states follows a Gaussian kernel in parameter space, with the QFIM as the weight matrix (Eq. 4).
• The generalized quantum natural gradient with β=1/2 (Gβ with β=1/2) is intrinsically stable without extra regularization (Eq. 6 discussion).
• Adaptive learning rates based on the kernel improve convergence and can surpass standard optimizers across PQC types (Fig. 3–5).
• A-QNG (β=1, with regularization) and A-GQNG (β=1/2, no regularization) yield order-of-magnitude reductions in infidelity for VQAs compared to Adam/LBFGS baselines (Fig. 5).
• The gradient variance has a lower bound independent of the number of qubits when the initial fidelity is bounded below (Eq. 13).
• The methods also effectively optimize driving protocols in quantum control problems, outperforming LBFGS in iteration efficiency (Fig. 5).