Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits

TL;DR

This paper proves that the choice of cost function in variational quantum algorithms critically affects trainability, with global observables causing barren plateaus even in shallow circuits, while local observables maintain trainability with logarithmic depth.

Contribution

It provides rigorous theoretical results linking cost function locality to gradient scaling and trainability in shallow parametrized quantum circuits.

Findings

Global observable costs lead to exponential vanishing gradients.

Local observable costs result in polynomially vanishing gradients for shallow circuits.

Simulations up to 100 qubits support theoretical predictions.

Abstract

Variational quantum algorithms (VQAs) optimize the parameters $\vec{\theta}$ of a parametrized quantum circuit $V(\vec{\theta})$ to minimize a cost function $C$. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming $V(\vec{\theta})$ is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining $C$ in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when $V(\vec{\theta})$ is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining $C$ with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of $V(\vec{\theta})$ is $\mathcal{O}(\log n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.