Analyzing the quantum approximate optimization algorithm: ansätze, symmetries, and Lie algebras

TL;DR

This work develops a comprehensive symmetry-and-Lie-algebra framework to analyze three QAOA maxcut ansätze on connected graphs. It delivers a complete Lie-algebra classification for the multi-angle (free) ansatz—falling into six families with exponential growth in most graphs—indicating prone barren plateaus, even at shallow depths. For the standard and orbit ansätze, it provides upper bounds via a natural symmetry-aligned Lie algebra and reveals hidden symmetries that complicate full classification, while proving exact results for path and cycle graphs and polynomial bounds for complete graphs. A central contribution is the characterization of invariant subspaces via representation theory and explicit character formulas, enabling a principled assessment of trainability and classical simulability across variational quantum algorithms. The framework highlights how graph symmetries and problem encodings shape the expressive power and optimization landscapes of QAOA, and it lays groundwork for applying similar analyses to broader variational circuits.

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 ansätze 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 ansätze. 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 ansätze 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.

Figures (17)

• Figure 1: Maxcut, QAOA and ansätze. (a) Given a graph, the maxcut problem is to determine a partition of the vertices into two complementary sets, such that the number of edges between those sets is as large as possible. (b) The QAOA algorithm is a hybrid quantum-classical algorithm that can be used to approximately solve the maxcut problem. The success of QAOA hinges on the ability to optimize the parametrized quantum circuit $U(\vec{\theta})$. Crucially, the trainability of QAOA can be linked to certain algebraic properties of $U(\vec{\theta})$. (c) Here we depict two of the considered QAOA ansätze: the standard and the multi-angle (or free) ansätze [see, e.g., Figs. \ref{['fig:ex:house']} and \ref{['fig:ex:house:orbit']}]. In the image, a gate with $ZZ$ indicates a two-qubit entangling gate generated by a $Z_w Z_{\tilde{w}}$ interaction, while the $X$ gate indicates a single-qubit rotation around the $x$-axis. Boxes with the same color share the same parameter. Hence, each gate in the free ansatz is individually parametrized and generated by a single Pauli operator, which makes its Lie algebra tractable across all graphs. In the standard case, all single-qubit gates share the same parameter, and similarly for all two-qubit gates. This means that the infinitesimal generators of the circuit are given by a sum of Paulis, which limits the ability to treat these cases.
• Figure 2: Example of house graph. Generators for the (a) standard and (b) the free ansatz.
• Figure 3: Gradient concentration for a single layer of the multi-angle QAOA applied to certain regular graphs. For each number of vertices $\abs{V}=n$, we pick a random $3$-regular or a complete graph. The variance of the gradients is obtained by sampling $100$ parameter values. The cost function $C(\vec{\theta})$ has been renormalized so that its values lie in $[-1,1]$. Refer also to the discussion in Sec. \ref{['sec:discussion']}.
• Figure 4: Symmetries in the standard-ansatz QAOA. The free-mixer symmetries $I^{\otimes n}$ and $X^{\otimes n}$ are complemented by symmetries arising from graph automorphism such as the permutation $(23)(45)$ for the house graph. These symmetries naturally act on quantum states (see text).
• Figure 5: Symmetry decomposition of the standard-ansatz QAOA for the house graph. (a) graph, automorphisms, symmetries; (b) invariant subspaces $\mathcal{H}_j$ as in Table \ref{['table:house:graph']} and projectors $P_j$ which are explicitly shown in Fig. \ref{['fig:matrices:house']} in App. \ref{['app:projections']}. This includes their dimension, their respective position in the $+1$ and the $-1$ eigenspace of $X^{\otimes 5}$ (denoted by $+$ or $-$), and explicit one-dimensional projectors. (c) natural and hidden symmetries; (d) transformation from $\mathcal{B}_{\text{inv}}$ in (b) to $\mathcal{B}_{\text{ext}}$ in (c), $\mathcal{B}_{\text{ext}}$ is not unique as shown by the red, nonzero entries in the last two rows; (e) maximum cut vectors and their (nonzero) support in the invariant subspaces.

Theorems & Definitions (96)

• Theorem 1: Free-mixer Lie algebras
• Definition 1: Archetypal graph
• Lemma 1
• proof
• Corollary 1
• Corollary 2
• Proposition 2: Lie-algebra and commutant hierarchy
• proof : Proof of Proposition \ref{['thm_dla_chain']}
• Proposition 3
• proof