Phase-Associative Memory: Sequence Modeling in Complex Hilbert Space

Abstract

We present Phase-Associative Memory (PAM), a recurrent sequence model in which all representations are complex-valued, associations accumulate in a matrix state $S_{t}$ $\in$ $\mathbb{C}^{d \times d}$ via outer products, and retrieval operates through the conjugate inner product $K_t^* \cdot Q_t / \sqrt{d}$. At $\sim$100M parameters on WikiText-103, PAM reaches validation perplexity 30.0, within $\sim$10\% of a matched transformer (27.1) trained under identical conditions, despite $4\times$ arithmetic overhead from complex computation and no custom kernels. We trace the experimental path from vector-state models, where holographic binding fails due to the $O(1/\sqrt{n})$ capacity degradation of superposed associations, to the matrix state that resolves it. The competitiveness of an architecture whose native operations are complex-valued superposition and conjugate retrieval is consistent with recent empirical evidence that semantic interpretation in both humans and large language models exhibits non-classical contextuality, and we discuss what this implies for the choice of computational formalism in language modeling.