Lie groups for quantum complexity and barren plateau theory

TL;DR

The paper surveys how Lie group theory, especially the geometry of $SU(2^n)$, provides a unifying lens for two key quantum computing challenges: a geometric theory of quantum complexity and the trainability of variational circuits. It develops a right-invariant, Pauli-symmetric Finsler metric on $SU(2^n)$ and derives Pauli geodesics for commuting Pauli generators, using Hamiltonian and Pauli coordinate representations linked by the exponential map. By exploiting Pauli symmetry and stabilizer subgroups, it shows how geodesics can be explicitly constructed in simple cases and outlines the role of the DLA in shaping circuit geometry. In the VQA context, the work connects $rak{g}$-purity and the dimension of the Dynamical Lie Algebra to gradient variance, explaining barren plateaus via algebraic structure and generalized entanglement/locality, with practical implications for circuit design and trainability.

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

Figures (2)

• Figure 1: The velocity $v(\lambda)$ of the curve $U$ at the point $U(\lambda)$.
• Figure 2: The reference frame $S$ and $S^{'}$.