On the dynamical Lie algebras of quantum approximate optimization algorithms

TL;DR

This paper analytically studies the dynamical Lie algebras of QAOA, providing bounds and explicit bases for specific graphs, and demonstrating the absence of barren plateaus in certain cases, advancing understanding of QAOA's expressiveness and trainability.

Contribution

It offers the first analytical characterization of DLAs for QAOA on specific graphs, including explicit bases and variance formulas, improving theoretical understanding.

Findings

Explicit basis for cycle graph DLA and its decomposition.

Proof of absence of barren plateaus for cycle graphs.

Dimension bounds and explicit bases for complete graph DLAs.

Abstract

Dynamical Lie algebras (DLAs) have emerged as a valuable tool in the study of parameterized quantum circuits, helping to characterize both their expressiveness and trainability. In particular, the absence or presence of barren plateaus (BPs) -- flat regions in parameter space that prevent the efficient training of variational quantum algorithms -- has recently been shown to be intimately related to quantities derived from the associated DLA. In this work, we investigate DLAs for the quantum approximate optimization algorithm (QAOA), one of the most studied variational quantum algorithms for solving graph MaxCut and other combinatorial optimization problems. While DLAs for QAOA circuits have been studied before, existing results have either been based on numerical evidence, or else correspond to DLA generators specifically chosen to be universal for quantum computation on a subspace of states. We initiate an analytical study of barren plateaus and other statistics of QAOA algorithms, and give bounds on the dimensions of the corresponding DLAs and their centers for general graphs. We then focus on the $n$-vertex cycle and complete graphs. For the cycle graph we give an explicit basis, identify its decomposition into the direct sum of a $2$-dimensional center and a semisimple component isomorphic to $n-1$ copies of $su(2)$. We give an explicit basis for this isomorphism, and a closed-form expression for the variance of the cost function, proving the absence of BPs. For the complete graph we prove that the dimension of the DLA is $O(n^3)$ and give an explicit basis for the DLA.