Effect of barren plateaus on gradient-free optimization

TL;DR

This paper demonstrates that gradient-free optimization methods are also hindered by barren plateaus in quantum algorithms, as their decision metrics become exponentially suppressed, requiring exponential resources for progress.

Contribution

It proves that gradient-free optimizers cannot overcome barren plateaus due to exponential suppression of cost function differences, challenging previous assumptions.

Findings

Gradient-free optimizers fail in barren plateaus due to exponential suppression.

Number of shots needed grows exponentially with qubits in barren landscapes.

Numerical experiments confirm exponential resource requirements for gradient-free methods.

Abstract

Barren plateau landscapes correspond to gradients that vanish exponentially in the number of qubits. Such landscapes have been demonstrated for variational quantum algorithms and quantum neural networks with either deep circuits or global cost functions. For obvious reasons, it is expected that gradient-based optimizers will be significantly affected by barren plateaus. However, whether or not gradient-free optimizers are impacted is a topic of debate, with some arguing that gradient-free approaches are unaffected by barren plateaus. Here we show that, indeed, gradient-free optimizers do not solve the barren plateau problem. Our main result proves that cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau. Hence, without exponential precision, gradient-free optimizers will not make progress in the optimization. We numerically confirm this by training in a barren plateau with several gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show that the numbers of shots required in the optimization grows exponentially with the number of qubits.