Effect of barren plateaus on gradient-free optimization

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.

The Optimization Landscape of Hybrid Quantum-Classical Algorithms: from Quantum Control to NISQ Applications

“…[51], it was shown that Newton-type algorithms make no difference because exponentially many measurements are still required to evaluate the Hessian matrix obtained by applying the parameter shift rule twice. In the presence of BPs, the landscape value will exhibit an exponential concentration about the mean [49,52], i.e., the variation Var θ [J(θ)] exponentially decays with the qubit number n. Other gradient-free optimizers (e.g., Nelder-Mead, Powell, and COBYLA) cannot improve the efficiency of optimization, because decisions made in these algorithms are based on the comparison of the objective function values between different points [52].…”

Mitigating Barren Plateaus with Transfer-learning-inspired Parameter Initializations

“…This means that when there establishes BP, the probability of finding parameters whose cost function is lower than the average value at a constant c is also exponentially suppressed. This exponential suppression of the cost difference makes the parameter optimization also difficult with gradient-free methods [44], where the parameter update is based on the sampled cost difference. It has been shown that BP is equivalent to the cost concentration.…”

Policy Gradient Approach to Compilation of Variational Quantum Circuits

“…In a barren plateau, ∇ θ F → 0 [22,24] and the value of the cost function is exponentially flattened in the number of qubits. In our case, we reformulate the problem in such a way that the space of solutions is no longer a set of possible configurations, but rather a space of (Gaussian) distributions.…”

“…This phenomenon, which exponentially increases the resources required to train large scale quantum neural networks, has been demonstrated in a number of proposed architectures and classes of cost functions [22,23]. Even gradient-free optimizers do not solve the barren plateau problem, as cost function differences, which are the basis for making decisions in a gradient-free optimization, are exponentially suppressed in a barren plateau [24].…”