Lie groups for quantum complexity and barren plateau theory

TL;DR

This paper reviews how Lie groups and their algebras provide a mathematical framework to analyze quantum computational complexity and the barren plateau problem in variational quantum algorithms, offering geometric and algebraic insights.

Contribution

It introduces Lie group theory to analyze quantum complexity and barren plateaus, connecting geometric and algebraic approaches to these fundamental issues.

Findings

Lie groups describe quantum complexity via shortest paths on $SU(2^n)$.

Dynamical Lie Algebra identifies sources of untrainability in VQAs.

Framework unifies geometric and algebraic analyses of quantum problems.

Abstract

Advances in quantum computing over the last two decades have required sophisticated mathematical frameworks to deepen the understanding of quantum algorithms. In this review, we introduce the theory of Lie groups and their algebras to analyze two fundamental problems in quantum computing as done in some recent works. Firstly, we describe the geometric formulation of quantum computational complexity, given by the length of the shortest path on the $SU(2^n)$ manifold with respect to a right-invariant Finsler metric. Secondly, we deal with the barren plateau phenomenon in Variational Quantum Algorithms (VQAs), where we use the Dynamical Lie Algebra (DLA) to identify algebraic sources of untrainability