From Barren Plateaus to SPSA Optimization in Variational Quantum Eigensolvers

This paper investigates how the barren plateau (BP) phenomenon—where objective function gradients vanish exponentially as system size increases—affects the practical performance of the Simultaneous Perturbation Stochastic Approximation (SPSA) algorithm in Variational Quantum Eigensolvers (VQEs). Specifically, it seeks to quantify the impact of BP on optimization dynamics and the total measurement budget required when using finite-shot measurements.

The author develops a theoretical framework to analyze SPSA under finite-shot measurement noise. The approach involves deriving non-asymptotic characterizations of the bias and variance of the SPSA gradient estimator, introducing a signal-to-noise ratio (SNR) metric to assess gradient reliability, and establishing convergence guarantees for the optimization trajectory. The study explicitly models the interplay between the intrinsic geometric properties of the VQE landscape and the statistical fluctuations introduced by finite-shot sampling.

The study demonstrates that the exponential decay of gradient energy associated with barren plateaus directly translates into an exponential increase in the computational cost of VQE optimization. The analysis reveals that maintaining a fixed relative optimization accuracy requires an exponentially growing number of iterations and an even larger total measurement budget, scaling with an exponent of 5/2 relative to the inverse of the gradient energy. This confirms that barren plateaus not only flatten the landscape but also fundamentally degrade the efficiency of stochastic optimization by necessitating significantly more measurement resources to distinguish the true gradient signal from statistical noise.