QAOA-MaxCut has barren plateaus for almost all graphs
Rui Mao, Pei Yuan, Jonathan Allcock, Shengyu Zhang
TL;DR
The paper investigates the dynamical Lie algebra (DLA) of QAOA-MaxCut for both weighted and unweighted graphs to understand the trainability of variational quantum algorithms and the emergence of barren plateaus. It develops a splitting-lemma framework and a practical algorithm to compute DLAs efficiently, using spectral properties of the operator $f$ to certify when the DLA is free (i.e., equals the multi-angle DLA) or exponentially large. The authors prove that for most graphs, including Erdős–Rényi random graphs, the DLA has dimension $\Theta(4^n)$, causing loss-variance $O(1/2^n)$ and hence barren plateaus, and they also construct explicit graph families with free DLAs. They validate the theory with numerical results on small graphs and the MQLib MaxCut benchmark, showing that a large fraction of instances possess massively large DLAs, while they also identify graph constructions and extensions that preserve freeness. Overall, the work clarifies why QAOA-MaxCut faces trainability challenges on typical graphs and provides a pathway for DLA-aware design and “DLA-engineering” to exploit or mitigate these effects in practice.
Abstract
The QAOA has been the subject of intense study over recent years, yet the corresponding Dynamical Lie Algebra (DLA)--a key indicator of the expressivity and trainability of VQAs--remains poorly understood beyond highly symmetric instances. An exponentially scaling DLA dimension is associated with the presence of so-called barren plateaus (BP) in the optimization landscape, which renders training intractable. In this work, we investigate the DLA of QAOA applied to the canonical MaxCut, for both weighted and unweighted graphs. For weighted graphs, we show that when the weights are drawn from a continuous distribution, the DLA dimension grows as $Θ(4^n)$ almost surely for all connected graphs except paths and cycles. In the more common unweighted setting, we show that asymptotically all but an exponentially vanishing fraction of graphs have $Θ(4^n)$ large DLA dimension. The entire simple Lie algebra decomposition of the corresponding DLAs is also identified, from which we prove that the variance of the loss function is $O(1/2^n)$, implying that QAOA on these weighted and unweighted graphs all suffers from BP. Moreover, we give explicit constructions for families of graphs whose DLAs have exponential dimension, including cases whose MaxCut is in $\mathsf P$. Our proof of the unweighted case is based on a number of splitting lemmas and DLA-freeness conditions that allow one to convert prohibitively complicated Lie algebraic problems into amenable graph theoretic problems. These form the basis for a new algorithm that computes such DLAs orders of magnitude faster than previous methods, reducing runtimes from days to seconds on standard hardware. We apply this algorithm to MQLib, a classical MaxCut benchmark suite covering over 3,500 instances with up to 53,130 vertices, and find that, ignoring edge weights, at least 75% of the instances possess a DLA of dimension at least $2^{128}$.