On the dynamical Lie algebras of quantum approximate optimization algorithms

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.

💡 Research Summary

This paper investigates the dynamical Lie algebras (DLAs) associated with the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut, linking DLA structure to the presence or absence of barren plateaus (BPs) in variational quantum algorithms (VQAs). The authors first establish general upper bounds on the dimension of the DLA and its center for arbitrary graphs, exploiting graph automorphism symmetries. They then focus on two highly symmetric families: the n‑vertex cycle graph Cₙ and the complete graph Kₙ.

For Cₙ, they prove that the DLA decomposes as a direct sum of a 2‑dimensional center and a semisimple component isomorphic to (n‑1) copies of su(2), giving a total semisimple dimension of 3(n‑1). An explicit basis for each su(2) copy is constructed, allowing a closed‑form calculation of the loss‑function variance. The variance remains Θ(1) with respect to n, demonstrating that BPs do not occur for QAOA on cycle graphs despite the relatively low DLA dimension.

For Kₙ, the authors determine the exact DLA dimension to be Θ(n³) and provide an explicit generating set. Although this dimension is far smaller than the full su(2ⁿ) (which scales as 4ⁿ), it is still polynomial, indicating that the circuit is not universal but retains enough complexity to make classical simulation difficult. The center of the DLA for Kₙ is larger than for Cₙ, reflecting the richer symmetry (the full symmetric group Sₙ).

The paper connects these structural results to the variance formula derived in earlier works:
Var_θ