Quantum Computing’s ‘barren Plateaus’ Overcome with Extra Circuit Parameters
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Variational Quantum Algorithms (VQAs) represent a promising avenue for near-term quantum computing, but optimisation performance can be severely limited by fundamental issues such as barren plateaus and overparametrisation. Himuro Hashimoto, Akio Nakabayashi, and Lento Nagano, from Yokogawa Electric Corporation and The University of Tokyo, with colleagues including Yutaro Iiyama, Ryu Sawada, and Junichi Tanaka, present a comprehensive numerical study investigating both barren plateaus and overparametrisation within VQA optimisation. Their work significantly advances understanding by quantitatively evaluating the impacts of these phenomena, and their interplay, using a one-dimensional transverse and longitudinal field Ising model.
This research offers a crucial framework for designing more effective VQAs and ansatzes, providing theoretical support for parameter optimisation behaviours in practical applications. These algorithms, reliant on parametrised quantum circuits, frequently encounter difficulties stemming from vanishing gradients known as barren plateaus and the presence of undesirable local minima within the cost function landscape. Numerical investigations have indicated that increasing the number of parameters in a circuit, a state termed overparametrisation, can significantly reduce the likelihood of becoming trapped in these local minima. While theoretical understanding of both barren plateaus and overparametrisation has progressed, a comprehensive study examining both phenomena concurrently has remained elusive. This work presents a detailed numerical analysis of VQAs, quantitatively assessing the impacts of barren plateaus and overparametrisation, and their combined influence on circuit optimisation. The research employs concrete implementations of the one-dimensional transverse and longitudinal field Ising model to evaluate performance. Numerical results are then carefully compared against established theoretical diagnostics for both barren plateaus and overparametrisation, providing a crucial validation of existing models. The framework developed in this study aims to establish a guiding principle for the design of VQAs and associated ansatzes, offering theoretical justification for parameter optimisation behaviours in practical applications. Specifically, the study investigates the convergence of the VQE algorithm, focusing on the interplay between the number of ansatz layers and training iterations. Results are presented as a two-dimensional plane illustrating energy accuracy, revealing three distinct regions characterised by differing convergence behaviours. For a sufficiently large number of iterations, the boundaries between these regions align with predictions based on the quantum Fisher information matrix and the variance of the gradient, corroborating previous research. Furthermore, the observed exponential convergence of energy accuracy in the overparametrisation regime aligns with existing findings. Intriguingly, the research also reveals a non-monotonic relationship between the number of iterations required for convergence and the number of layers in the circuit, suggesting a more complex interplay between these parameters than previously understood. This motivates further theoretical investigation into estimating the optimal number of parameters and iterations needed to achieve robust convergence in VQA optimisation. The comprehensive visualisation of training output presented in this work is intended to serve as a valuable resource for both experts and newcomers, providing insights into how circuit structure and hyperparameter choices impact variational optimisation. Numerical optimisation of transverse field Ising models using a 72-qubit processor demonstrates promising results A 72-qubit superconducting processor forms the foundation of this study, which investigates the interplay between barren plateaus and overparametrization in variational quantum algorithms. Researchers employed a numerical approach to comprehensively evaluate the impacts of these phenomena on circuit optimization, utilising concrete implementations of a one-dimensional transverse and longitudinal field Ising model. The optimisation process began with a parameter selection strategy where, at each epoch t, a random parameter was chosen, excluding the final parameter from the previous epoch to prevent redundant optimisation steps. Cost function values were recorded at the last step of each epoch to track optimisation progress. The core of the methodology involved performing the ERNFT optimisation 30times, each run initialised with distinct random parameters drawn from a uniform distribution between -π and π for RY and RZ gates. Average values of the relative residual energy E(θ) were then calculated across these 30 runs at each epoch for each experimental setting. All simulations were conducted using the noiseless statevector simulator within the Qulacs Python library, with optimisers also implemented in Python. This setup allowed for a rigorous and reproducible assessment of the optimisation landscape. Heat maps were generated to visualise the dependence of relative residual energy on both the number of ansatz layers L and the number of epochs t for system sizes N ranging from 4 to 10. These maps were demarcated by dotted and dashed lines, representing the point at which the number of parameters reached the maximal rank of the quantum Fisher information matrix (QFIM) and the threshold where the normalised variance of the gradient fell below a value vth, respectively. Three distinct regions emerged, categorised by relative residual energy, providing insights into the expressibility of the ansatz, gradient behaviour, and the onset of overparametrization. Further analysis focused on the gradient of the cost function, specifically investigating the variance of the gradients Var(∂kE(θ)), which is indicative of barren plateaus. The variance was calculated from 10,000 randomly generated parameter sets for each combination of N and L, revealing an exponential decrease with increasing qubit number. This detailed examination of gradient behaviour, alongside the QFIM rank and relative residual energy, enabled a thorough interpretation of the numerical results from a theoretical perspective. VQE performance stratification via layer depth and optimisation epochs reveals key trends in circuit scalability Numerical studies reveal distinct regions in Variational Quantum Algorithm (VQA) performance based on circuit layers and training iterations. Specifically, energy accuracy is quantified across a two-dimensional plane defined by the number of VQE ansatz layers and the number of epochs, demonstrating three quantitatively different regions. For a sufficiently large number of iterations, the boundaries between these regions correlate with the maximal rank of the quantum Fisher information matrix and the variance of the gradient of the loss function, aligning with previous investigations. Exponential convergence of energy accuracy is observed in the overparametrization regime, consistent with results reported in several studies. Interestingly, the number of iterations required for converging energy accuracy varies non-monotonically with the number of layers in the circuit within a specific region. This observation motivates further theoretical work to estimate the optimal number of parameters and iterations needed for effective convergence. The research provides a comprehensive framework for evaluating the impacts of barren plateaus and overparametrization, and their interplay, on the optimization of a variational quantum circuit. This work utilizes concrete implementations of the one-dimensional transverse and longitudinal field Ising model to quantitatively assess VQA optimization. The framework presented aims to guide the design of VQA algorithms and ansatzes, providing theoretical support for parameter optimization behaviours in practical settings. By visualizing training output, this study offers insights into how circuit structure and hyperparameters affect variational optimization, potentially aiding broader understanding and application of VQA techniques. Layer depth, optimisation iterations and qubit count influence variational quantum algorithm performance significantly Researchers have performed a comprehensive numerical study of variational quantum algorithms, benchmarking ansatz structures, optimizers and hyperparameters to assess trainability. The investigation focused on the transverse and longitudinal field Ising model, quantitatively evaluating the impacts of barren plateaus and overparametrization on circuit optimization.
Results demonstrate a relationship between the number of layers in a quantum circuit, the number of iterations during optimization, and the number of qubits, revealing how these factors influence energy accuracy. Specifically, the study found that residual energy increases when the number of layers exceeds a certain threshold due to barren plateau effects, but decreases again with further increases in layers due to overparametrization. The observed behaviour aligns with existing theoretical understanding of both phenomena, and the framework established in this work offers a means of quantifying trainability for new variational algorithms. Furthermore, a power-law decay in residual energy was observed as a function of optimization iterations near a specific point, suggesting a potentially important area for further investigation. The authors acknowledge that all currently known barren-plateau-free ansatzes are potentially classically simulatable. However, they propose that numerical studies using this framework, even within practical settings, may guide efforts to achieve quantum advantage and provide insights into universal convergence behaviours. Future research directions include a deeper theoretical understanding of the decay rates observed with increasing layers, and continued exploration of this framework to select hyperparameters that improve accuracy and potentially circumvent limitations related to classical simulability. 👉 More information 🗞 Comprehensive Numerical Studies of Barren Plateau and Overparametrization in Variational Quantum Algorithm 🧠 ArXiv: https://arxiv.org/abs/2602.03291 Tags: