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

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.

💡 Research Summary

This paper presents a systematic numerical investigation of two central challenges in variational quantum algorithms (VQAs): barren plateaus (BPs) and over‑parametrization (OP). Using the one‑dimensional transverse‑and‑longitudinal field Ising model (TLFIM) as a concrete benchmark, the authors study how the depth of a hardware‑efficient ansatz (HEA) and the number of training epochs jointly affect the convergence of the variational quantum eigensolver (VQE) to the ground‑state energy.

The Hamiltonian is defined on a periodic chain with coupling J = 1 and transverse/longitudinal fields hX = hZ = 1. The ansatz consists of L layers, each containing single‑qubit RY and RZ rotations on every qubit followed by a nearest‑neighbour CNOT entangling block. The total number of variational parameters is p = 2 N L, where N is the number of qubits (4 ≤ N ≤ 10, with an additional N = 9 example). By varying L from 3 up to 201, the study spans regimes from under‑expressive to heavily over‑parameterized circuits.

Optimization is performed with a sequential minimal optimization (SMO) scheme originally proposed by Nakajima, Fuji, and Todo (referred to as NFT). To avoid order‑dependent artefacts, the authors randomize the parameter‑update order at each epoch, defining an “epoch‑wise random NFT” (ERNFT). Each configuration is run 30 times with independent uniform random initial parameters (θ ∈