Quantum imaginary time evolution (QITE) algorithm is one of the most promising variational quantum algorithms (VQAs), bridging the current era of Noisy Intermediate-Scale Quantum devices and the future of fully fault-tolerant quantum computing. Although practical demonstrations of QITE and its potential advantages over the general VQA trained with vanilla gradient descent (GD) in certain tasks have been reported, a first-principle, theoretical understanding of QITE remains limited. Here, we aim to develop an analytic theory for the dynamics of QITE. First, we show that QITE can be interpreted as a form of a general VQA trained with Quantum Natural Gradient Descent (QNGD), where the inverse quantum Fisher information matrix serves as the learning-rate tensor. This equivalence is established not only at the level of gradient update rules, but also through the action principle: the variational principle can be directly connected to the geometric geodesic distance in the quantum Fisher information metric, up to an integration constant. Second, for wide quantum neural networks, we employ the quantum neural tangent kernel framework to construct an analytic model for QITE. We prove that QITE always converges faster than GD-based VQA, though this advantage is suppressed by the exponential growth of Hilbert space dimension. This helps explain certain experimental results in quantum computational chemistry. Our theory encompasses linear, quadratic, and more general loss functions. We validate the analytic results through numerical simulations. Our findings establish a theoretical foundation for QITE dynamics and provide analytic insights for the first-principle design of variational quantum algorithms.

翻译：量子虚时演化（QITE）算法是最具前景的变分量子算法（VQAs）之一，它连接了当前噪声中等规模量子器件时代与未来完全容错量子计算的发展。尽管已有研究报道了QITE的实际演示及其在某些任务中相较于采用普通梯度下降（GD）训练的通用VQA的潜在优势，但对QITE的第一性原理理论理解仍较为有限。本文旨在建立QITE动力学的解析理论。首先，我们证明QITE可被解释为一种采用量子自然梯度下降（QNGD）训练的通用VQA形式，其中逆量子费希尔信息矩阵充当学习率张量。这一等价性不仅建立在梯度更新规则层面，还通过作用量原理得以确立：变分原理可直接关联至量子费希尔信息度量中的几何测地线距离（至多相差一个积分常数）。其次，针对宽量子神经网络，我们运用量子神经正切核框架构建了QITE的解析模型。我们证明QITE始终比基于GD的VQA收敛更快，尽管这一优势会因希尔伯特空间维度的指数增长而受到抑制。这有助于解释量子计算化学中的某些实验结果。我们的理论涵盖线性、二次及更一般的损失函数。我们通过数值模拟验证了解析结果。本研究为QITE动力学奠定了理论基础，并为变分量子算法的第一性原理设计提供了解析视角。