Effect of barren plateaus on gradient-free optimization
This work addresses a debate in quantum computing regarding the effectiveness of gradient-free optimizers in the presence of barren plateaus, which is a critical problem for the development of scalable quantum algorithms.
This paper investigates the impact of barren plateaus on gradient-free optimization methods, demonstrating that these optimizers are also affected. The authors prove that cost function differences, crucial for gradient-free decisions, are exponentially suppressed in barren plateaus, leading to an exponential increase in required shots with the number of qubits for algorithms like Nelder-Mead, Powell, and COBYLA.
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.