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

TL;DR

Address the problem of VQA trainability by numerically analyzing BP and OP in a VQE setup for the 1D transverse/longitudinal field Ising model, validating how gradient variance and the maximal rank of the quantum Fisher information matrix (QFIM) delimit trainable regions. The approach combines a Hardware Efficient Ansatz (HEA) with epoch-wise random NFT optimization (ERNFT) and tracks the relative energy $E(\bm{\theta})$ across system size $N$, layer depth $L$, and training iterations $t$. Key contributions reveal three regimes in the $L$-$t$ plane (pre-BP, BP, OP), with OP yielding exponential convergence once $p=2NL$ exceeds the maximal QFIM rank. Practically, the results provide guidance for selecting ansatz depth and optimizer strategies to achieve robust convergence in near-term quantum devices, supported by a link between BP/OP behavior and frame potential diagnostics.

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.

Figures (13)

• Figure 1: Summary of numerical studies presented in this paper showing energy accuracy $E$ of VQE algorithm in the two-dimensional plane of the number of VQE ansatz layers ($L$) and the number of epochs ($t$). Detailed definitions of the presented quantities as well as the horizontal lines and the arrows are explained in the main text.
• Figure 2: Schematic picture of the spin system for $N=6$ with the periodic boundary condition. The transverse and longitudinal fields $h_X$ and $h_Z$ are applied at each site, and each neighboring pair has an Ising interaction with strength $J$.
• Figure 3: An example of HEA with $(N, L) = (6, 3)$ drawn using IBM Qiskit library qiskit2024. Each pair of parametrized block and entanglement block is repeated $L-1$ times, followed by another parametrized block.
• Figure 4: Heat maps of relative residual energy $E$ in the $t$-$L$ plane at $N=4$, 6, 8 and 10. As mentioned in Section \ref{['sec:misc settings']}, the $E$ value averaged over 30 runs is shown at each $(t, L)$ grid. The black dashed lines represent the locations where ${\mathrm{Var}(\partial_0 E)}_{\mathrm{norm}} = v_\mathrm{th}$, as discussed in Section \ref{['sec:BP']}. The white dotted lines correspond to the $L$ values where $p = \max(\mathrm{rank}(\mathrm{QFIM}))$, as discussed in Section \ref{['sec:OP']}.
• Figure 5: Variance $\mathrm{Var}(\partial_0 E)$ of partial gradients of relative residual energy with respect to the parameter $\theta_0$. The $\mathrm{Var}(\partial_0 E)$ is calculated from $10000$ random parameter sets at each point, and is shown as a function of $L$ ($N$) at fixed $N$ ($L$) values in the top (bottom) panel.