Rand-SEMI-QAOA: Finite-Budget Depth-One MaxCut Ensembles on Compressed Quantum Registers

Abstract

We introduce Rand-SEMI-QAOA, a finite-budget QRAO--QAOA ensemble for MaxCut based on a $(3,1)$-QRAC relaxation. The method samples labels uniformly without replacement from the product-$X$ family of an implemented Galois stabilizer mutually unbiased basis (MUB) system. Each selected state--mixer pair is optimized independently for relaxed QRAO energy and then evaluated by deterministic Pauli-sign decoding. Exhaustive scans of the implemented noncomputational MUB catalog place the product-$X$ family first in 18 of 20 validated family-mean cells and in every tested cell for $r=5,6,7$. On a complete cohort of $2{,}400$ random connected 3-regular MaxCut instances with $n\in\{18,20,22\}$, the capped matched-cardinality schedule attains a graph-mean decoded best-of-set approximation ratio of $0.9421$ with one QAOA layer, while the $K=r^2$ schedule attains $0.9319$. These statistics are conditional on the disclosed frozen selector pools and do not estimate variability over selector seeds. An exact gauge identity shows that product-family labels generate the orbit of the QRAO Hamiltonian under an $r$-dimensional sign-gauge group. For common angles and relaxed energy, deterministic syndrome analysis proves branchwise dephasing, while an anisotropic Gaussian surrogate controls the coherent label-averaged response on the scale $β=b/r$. The theory does not order independently optimized decoded maxima. The results are finite-size and resource-explicit and do not establish quantum advantage.