Analyzing the quantum approximate optimization algorithm: ans\"atze, symmetries, and Lie algebras

TL;DR

This paper investigates the algebraic structures of different QAOA ansätze for max-cut problems, revealing their symmetries, Lie algebras, and implications for barren plateaus and classical simulability.

Contribution

It fully characterizes the Lie algebras of multi-angle QAOA ansätze across connected graphs and introduces a symmetry framework for analyzing variational quantum algorithms.

Findings

Lie algebras of multi-angle ansatz fall into six families

Large Lie algebra dimensions imply susceptibility to barren plateaus

Symmetry analysis suggests potential classical simulability of QAOA

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) has been proposed as a method to obtain approximate solutions for combinatorial optimization tasks. In this work, we study the underlying algebraic properties of three QAOA ans\"atze for the maximum-cut (maxcut) problem on connected graphs, while focusing on the generated Lie algebras as well as their invariant subspaces. Specifically, we analyze the standard QAOA ansatz as well as the orbit and the multi-angle ans\"atze. We are able to fully characterize the Lie algebras of the multi-angle ansatz across arbitrary connected graphs, finding that they only fall into one of just six families. Besides the cycle and the path graphs, the Lie dimensions for every graph are exponentially large in the system size, meaning that multi-angle ans\"atze are extremely prone to exhibiting barren plateaus. Then, a similar quasi-graph-independent Lie-algebraic characterization beyond the multi-angle ansatz is impeded as the circuit exhibits additional "hidden" symmetries besides those naturally arising from a certain parity-superselection operator and all automorphisms of the considered graph. Disregarding the "hidden" symmetries, we can upper bound the dimensions of the orbit and the standard Lie algebras, and the dimensions of the associated invariant subspaces are determined via explicit character formulas. To finish, we conjecture that (for most graphs) the standard Lie algebras have only components that are either exponential or that grow, at most, polynomially with the system size. This would imply that the QAOA is either prone to barren plateaus, or classically simulable. More generally, our work provides a symmetry framework and tools to analyze any desired variational quantum algorithm.