Cost function dependent barren plateaus in shallow parametrized quantum circuits.

Area of Science:

• Quantum Computing
• Quantum Machine Learning

Background:

• Variational quantum algorithms (VQAs) are heuristic methods for noisy quantum computers.
• Their scaling properties and trainability challenges, like barren plateaus, are not fully understood.

Purpose of the Study:

• To rigorously analyze the trainability of VQAs based on circuit structure and observable choice.
• To identify conditions that avoid barren plateaus and ensure efficient optimization.

Main Methods:

• Theoretical analysis of parametrized quantum circuits V(θ) using alternating layered ansatze.
• Proving results on gradient vanishing based on the choice of cost function observables (global vs. local).
• Large-scale simulations of a quantum autoencoder up to 100 qubits.

Main Results:

• Global observables in VQAs lead to exponentially vanishing gradients (barren plateaus), even for shallow circuits.
• Local observables result in at worst polynomially vanishing gradients for circuits of depth [Formula: see text].
• A direct link between the locality of observables and VQA trainability is established.

Conclusions:

• VQAs using global observables require revised cost function designs to ensure trainability.
• Local observables offer a path to overcome barren plateaus and improve VQA performance.
• The findings are crucial for developing scalable and practical quantum algorithms.