Comprehensive Numerical Studies of Barren Plateau and Overparametrization in Variational Quantum Algorithm

TL;DR

This paper provides a comprehensive numerical analysis of the barren plateau problem and overparametrization effects in variational quantum algorithms, focusing on their interplay and impact on optimization performance.

Contribution

It offers the first detailed numerical study of both barren plateau and overparametrization phenomena within the same framework for VQAs, supported by theoretical diagnostics.

Findings

Overparametrization reduces local minima traps.

Barren plateau causes vanishing gradients, hindering optimization.

Numerical results align with theoretical predictions.

Abstract

The variational quantum algorithm (VQA) with a parametrized quantum circuit is widely applicable to near-term quantum computing, but its fundamental issues that limit optimization performance have been reported in the literature. For example, VQA optimization often suffers from vanishing gradients called barren plateau (BP) and the presence of local minima in the landscape of the cost function. Numerical studies have shown that the trap in local minima is significantly reduced when the circuit is overparametrized (OP), where the number of parameters exceeds a certain threshold. Theoretical understanding of the BP and OP phenomena has advanced over the past years, however, comprehensive studies of both effects in the same setting are not fully covered in the literature. In this paper, we perform a comprehensive numerical study in VQA, quantitatively evaluating the impacts of BP and OP and their interplay on the optimization of a variational quantum circuit, using concrete implementations of one-dimensional transverse and longitudinal field quantum Ising model. The numerical results are compared with the theoretical diagnostics of BP and OP phenomena. The framework presented in this paper will provide a guiding principle for designing VQA algorithms and ansatzes with theoretical support for behaviors of parameter optimization in practical settings.