Benchmarking Lie-Algebraic Pretraining and Non-Variational QWOA for the MaxCut Problem

TL;DR

This work compares two trains-on-training strategies for MaxCut on small graphs: Lie algebraic pretraining (LAP), which uses a dynamically generated Lie algebra to pretrain a large PQC, and non-variational QWOA (NV-QWOA), which constrains the trainable space to three hyperparameters. On 16-vertex graphs ($n=16$) with depth $p=256$, NV-QWOA achieves near-optimal performance remarkably quickly, while LAP improves over random initialization but remains slower and, in this benchmark, outperformed by NV-QWOA. The study also analyzes the behavior of each approach across 400 graphs (ER and $3$-regular) and discusses instances where LAP fails to beat random initializations and where NV-QWOA consistently converges, highlighting the practical robustness of the NV-QWOA scheme. Overall, the results suggest that a highly structured, low-dimensional parameterization can offer superior trainability and reliable performance on near-term quantum devices, motivating further scaling and generalization studies.

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a leading candidate for achieving quantum advantage in combinatorial optimization on Near-Term Intermediate-Scale Quantum (NISQ) devices. However, random initialization of the variational parameters typically leads to vanishing gradients, rendering standard variational optimization ineffective. This paper provides a comparative performance analysis of two distinct strategies designed to improve trainability: Lie algebraic pretraining framework that uses Lie-algebraic classical simulation to find near-optimal initializations, and non-variational QWOA (NV-QWOA) that targets a restrict parameter subspace covered by 3 hyperparameters. We benchmark both methods on the unweighted Maxcut problem using a circuit depth of $p = 256$ across 200 Erdős-Rényi and 200 3-regular graphs, each with 16 vertices. Both approaches significantly improve upon the standard randomly initialized QWOA. NV-QWOA attains a mean approximation ratio of 98.9\% in just 60 iterations, while the Lie-algebraic pretrained QWOA improves to 77.71\% after 500 iterations. That optimization proceeds more quickly for NV-QWOA is not surprising given its significantly smaller parameter space, however, that an algorithm with so few tunable parameters reliably finds near-optimal solutions is remarkable. These findings suggest that the structured parameterization of NV-QWOA offers a more robust training approach than pretraining on lower-dimensional auxiliary problems. Future work is needed to confirm scaling to larger problem sizes and to asses generalization to other problem classes.

Figures (4)

• Figure 1: Examples of common graph structures.
• Figure 2: Initial parameters for QWOA for Maxcut with 256 layers ($p=256$). (A) Example solution parameters for the 16-vertex path graph, used as initial parameters for Lie algebraic pretraining. (B) Example parameters for NV-QWOA, namely $(\beta,\gamma,t) = (0.35,5.3,4)$.
• Figure 3: Individuals runs of the same problem sets on three different algorithms.
• Figure 4: Mean approximation ratio (with 95% confidence interval) of each iteration across runs of the same problem sets on the three algorithms.