An Analytic Theory of Quantum Imaginary Time Evolution

TL;DR

This paper develops an analytic theory for Quantum Imaginary Time Evolution (QITE), revealing its connection to Quantum Natural Gradient Descent and providing insights into its convergence properties and geometric interpretation.

Contribution

It introduces a first-principles theoretical framework for QITE, linking it to quantum Fisher information geometry and neural tangent kernel methods, and compares its convergence to gradient descent.

Findings

QITE can be interpreted as a form of VQA trained with Quantum Natural Gradient Descent.

QITE always converges faster than GD-based VQA, with the advantage affected by Hilbert space size.

The theory is validated through numerical simulations and explains experimental results in quantum chemistry.

Abstract

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.