Characterizing barren plateaus in quantum ansätze with the adjoint representation.

Area of Science:

• Quantum Computing
• Quantum Machine Learning
• Theoretical Physics

Background:

• Variational quantum algorithms (VQAs) are heuristic methods for near-term quantum computers.
• Parameterized quantum circuits in VQAs are related to Lie groups, suggesting group properties influence algorithm behavior.
• Barren plateaus, characterized by vanishing gradients, are a major obstacle in VQA training, but lack a solid theoretical foundation.

Purpose of the Study:

• To develop a theoretical framework for understanding barren plateaus in VQAs.
• To connect the phenomenon of barren plateaus to the group structure of parameterized quantum circuits.
• To provide a method for calculating the gradient variance in VQAs.

Main Methods:

• Utilized tools from the representation theory of compact Lie groups.
• Formulated a theory of barren plateaus for parameterized quantum circuits.
• Focused on circuits where observables are within the dynamical Lie algebra.

Main Results:

• Developed a theory applicable to common ansätze like the Hamiltonian Variational Ansatz and Quantum Alternating Operator Ansatz.
• Established a method for computing the exact variance of the gradient of the cost function for quantum compound ansätze.
• Identified commonplace mixing conditions that lead to barren plateaus.

Conclusions:

• The study provides the first theoretical derivation for barren plateaus in VQAs.
• The developed theory offers insights into the behavior of various quantum machine learning models.
• This work paves the way for mitigating barren plateaus and improving VQA training.