SQUARE: Structured Quantum Representation Adapters as Compact Quadratic Feature Maps for Frozen Language Models

Emily Jimin Roh, Hyojun Ahn, Hoyeong Lee, Soohyun Park, Sung Whan Yoon +2

Frozen language models (LMs) are increasingly used as fixed feature extractors for downstream reranking, scoring, and preference modeling, raising a practical question: how should a compact module represent interactions among features in a fixed low-dimensional bottleneck? Common linear and low-rank adapters remain linear at the adaptation module itself, whereas explicit second-order alternatives introduce pairwise interactions through direct parameterization or predefined factorizations. We propose SQUARE, a Structured QUAntum REpresentation adapter that amplitude-encodes the bottleneck vector, applies a parameterized quantum circuit, and measures the resulting state. We show that each basis-probability feature is exactly a normalized quadratic form in the bottleneck coordinates, while the additional Pauli-$Z$ readouts are signed linear combinations of these probabilities. The measured map can therefore parameterize interactions over $O(d^2)$ coordinate pairs through a small set of shared circuit parameters, where $d$ is the bottleneck dimension. It provides a structured parameterization within, rather than beyond, the classical normalized-quadratic feature class. In a disjoint…