Cost function dependent barren plateaus in shallow parametrized quantum circuits

TLDR

Variational quantum algorithms optimize circuit parameters to minimize a cost function, but their practical scalability remains unproven. The study rigorously proves two results for alternating layered ansatz circuits composed of local 2‑design blocks. The authors analyze these circuits and illustrate the results with large‑scale simulations of a quantum autoencoder up to 100 qubits. They show that cost functions based on global observables cause exponentially vanishing gradients even for shallow circuits, while local observables yield at most polynomially vanishing gradients when depth is O(log n), linking locality to trainability.

Abstract

Abstract Variational quantum algorithms (VQAs) optimize the parameters θ of a parametrized quantum circuit V ( θ ) 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 ( θ ) 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 ( θ ) 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 ( θ ) is $${\mathcal{O}}(\mathrm{log}\,n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>log</mml:mi> <mml:mspace/> <mml:mi>n</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . 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.