An Analytic Theory of Quantum Imaginary Time Evolution

TL;DR

This work provides a principled analytic framework for quantum imaginary time evolution (QITE) by establishing a first-principle equivalence to quantum natural gradient descent (QNGD) on the quantum Fisher information geometry. It develops a quantum neural tangent kernel (QNTK) based model to analyze QITE dynamics in wide quantum neural networks and demonstrates a convergence speedup over gradient-descent–based VQAs, with explicit relations between QITE and GD quantities and extensions to general loss functions. The theory unifies state- and parameter-space perspectives via an action-principle mapping to the quantum geometric tensor, and is validated by comprehensive numerical studies under Haar-random ensembles. The results offer analytic insight for designing variational quantum algorithms with principled convergence properties and help explain experimental observations in quantum chemistry simulations. Overall, the paper provides a first-principle foundation for QITE dynamics and paves the way for geometry-guided design of variational quantum algorithms.

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.

Figures (8)

• Figure 1: Overview of Main Results. Firstly, we uncover and establish a first-principle equivalence between QITE and QNGD-based VQAs by deriving that the objective functions, the variational principles and the dynamical equations are identical up to an integration constant with general loss function in continuous-time limit. Secondly, we focus on quadratic and linear loss, and leverage QNTK theory to derive a first-principle model to characterize the training dynamics of QITE in the regimes of interest.
• Figure 2: Training Dynamics of GD-based VQAs and QITE with quadratic loss function. Here, in the example of XXZ model, we respectively investigate the QNTK $K(t)$, the residual error $\epsilon(t)$, the average trace and the off-diagonal terms of the Fubini-study metric tensor $g$. Each numerical curves are plotted by averaging over 50 times, indicating 50 initializations. We adopt HEA ansatz with 6 layers, and set the number of qubits $n = 3$. The learning rate for optimization is $\eta = 0.001$ with 200 steps. Red curves (denoted as "Numerical QITE") represent ensemble average results of QITE. Blue curves (denoted as "Numerical GD") represent the ensemble average numerical results of GD-based VQAs. Green dashed curves represent the analytic prediction of the dynamics of QITE. We also plot the gray lines in the plot of average trace of $g$, indicating 50 random samples.
• Figure 3: Training Dynamics of QNNs in GD-based VQAs and QITE with linear loss function. Comparing with Fig \ref{['fig:Quadratic_results']}, we additionally investigate the relative dQNTK value $\lambda (t)$ which drives the dynamics of $K(t)$ and $\epsilon(t)$. Each numerical curves are plotted by averaging over 50 times, indicating 50 initializations. The settings is identical to Fig. \ref{['fig:Quadratic_results']}.
• Figure 4: The diagonal approximation (yellow circles) is highlighted at selected steps to evaluate its agreement with the full $K_\text{QITE}$. The agreement confirms that diagonal elements dominate the kernel structure, supporting the validity of the approximation in practice.
• Figure 5: The diagonal approximation (yellow circles) is highlighted at selected steps to evaluate its agreement with the full $K_\text{QITE}$. The agreement confirms that diagonal elements dominate the kernel structure, supporting the validity of the approximation in practice.

Theorems & Definitions (17)

• Theorem 2.1: The Objective Equivalence of QITE and QNGD-based VQAs in the Continuous-Time Limit
• Theorem 2.2: Variational Principle of QNGD-based VQAs with General Loss Function
• Theorem 2.3: Variational Principle Equivalence between QNGD-based VQAs and QITE with Linear Loss Function
• Theorem 2.4: Generalized McLachlan Variational Principle for QITE with general loss function
• Remark
• Theorem 2.5: Variational Principle Equivalence of QITE and QNGD-based VQAs with General Loss Function
• Definition 2.6: Quantum Neural Tangent Kernel (QNTK) for QITE
• Lemma 2.7
• Proposition 2.8
• Definition 2.9: Quantum Meta-Kernel for QITE