Variational Quantum Classifiers Through the Lens of the Hessian

TL;DR

This paper explores the curvature and loss landscape of variational quantum classifiers by calculating and visualizing the Hessian, providing insights into their trainability and convergence behavior on quantum hardware.

Contribution

It introduces the analysis of the Hessian matrix for variational quantum classifiers, linking curvature information to training dynamics and convergence in quantum machine learning.

Findings

Hessian analysis reveals the loss landscape topology of VQCs.

Eigenvalues of the Hessian relate to trainability and convergence.

Adaptive Hessian learning rate improves training efficiency.

Abstract

In quantum computing, the variational quantum algorithms (VQAs) are well suited for finding optimal combinations of things in specific applications ranging from chemistry all the way to finance. The training of VQAs with gradient descent optimization algorithm has shown a good convergence. At an early stage, the simulation of variational quantum circuits on noisy intermediate-scale quantum (NISQ) devices suffers from noisy outputs. Just like classical deep learning, it also suffers from vanishing gradient problems. It is a realistic goal to study the topology of loss landscape, to visualize the curvature information and trainability of these circuits in the existence of vanishing gradients. In this paper, we calculate the Hessian and visualize the loss landscape of variational quantum classifiers at different points in parameter space. The curvature information of variational quantum classifiers (VQC) is interpreted and the loss function's convergence is shown. It helps us better understand the behavior of variational quantum circuits to tackle optimization problems efficiently. We investigated the variational quantum classifiers via Hessian on quantum computers, starting with a simple 4-bit parity problem to gain insight into the practical behavior of Hessian, then thoroughly analyzed the behavior of Hessian's eigenvalues on training the variational quantum classifier for the Diabetes dataset. Finally, we show how the adaptive Hessian learning rate can influence the convergence while training the variational circuits.