# Phase-Associative Memory: Sequence Modeling in Complex Hilbert Space

arXiv: https://arxiv.org/abs/2604.05030 · HTML: https://mlsanity.com/paper/2604.05030/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 45 of its 113 references that are in the map.
- First posted on arXiv 6 Apr 2026.

## Builds on (most cited first, 30 of 45)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [Language Models are Few-Shot Learners](https://mlsanity.com/paper/2005.14165/) (2020)
- [Retrieval-Augmented Generation for Knowledge-Intensive NLP Tasks](https://mlsanity.com/paper/2005.11401/) (2020)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [RoFormer: Enhanced Transformer with Rotary Position Embedding](https://mlsanity.com/paper/2104.09864/) (2021)
- [Neural Machine Translation by Jointly Learning to Align and Translate](https://mlsanity.com/paper/1409.0473/) (2014)
- [Efficient Estimation of Word Representations in Vector Space](https://mlsanity.com/paper/1301.3781/) (2013)
- [Pointer Sentinel Mixture Models](https://mlsanity.com/paper/1609.07843/) (2016)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [Survey of Hallucination in Natural Language Generation](https://mlsanity.com/paper/2202.03629/) (2022)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Scaling LLM Test-Time Compute Optimally can be More Effective than Scaling Model Parameters](https://mlsanity.com/paper/2408.03314/) (2024)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [Not what you've signed up for: Compromising Real-World LLM-Integrated Applications with Indirect Prompt Injection](https://mlsanity.com/paper/2302.12173/) (2023)
- [Transformers are SSMs: Generalized Models and Efficient Algorithms Through Structured State Space Duality](https://mlsanity.com/paper/2405.21060/) (2024)
- [Ignore Previous Prompt: Attack Techniques For Language Models](https://mlsanity.com/paper/2211.09527/) (2022)
- [RWKV: Reinventing RNNs for the Transformer Era](https://mlsanity.com/paper/2305.13048/) (2023)
- [Retentive Network: A Successor to Transformer for Large Language Models](https://mlsanity.com/paper/2307.08621/) (2023)
- [Scaling and evaluating sparse autoencoders](https://mlsanity.com/paper/2406.04093/) (2024)
- [Gated Linear Attention Transformers with Hardware-Efficient Training](https://mlsanity.com/paper/2312.06635/) (2023)
- [Linear Transformers Are Secretly Fast Weight Programmers](https://mlsanity.com/paper/2102.11174/) (2021)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Parallelizing Linear Transformers with the Delta Rule over Sequence Length](https://mlsanity.com/paper/2406.06484/) (2024)
- [xLSTM: Extended Long Short-Term Memory](https://mlsanity.com/paper/2405.04517/) (2024)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Griffin: Mixing Gated Linear Recurrences with Local Attention for Efficient Language Models](https://mlsanity.com/paper/2402.19427/) (2024)
- [Dense Associative Memory for Pattern Recognition](https://mlsanity.com/paper/1606.01164/) (2016)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)

## Abstract (arXiv)

Experiments probing natural language processing by both humans and LLMs suggest that the meaning of a semantic expression is indeterminate prior to the act of interpretation rather than being specifiable simply as the sum of its parts (i.e. compositionality). This observer-dependent act dynamically actualizes meaning under genuine contextuality more consistent with quantum logical mechanisms than with classical Boolean approaches that assume separability, motivating an approach to language modeling that utilizes a Hilbert space formalism. In this work, we introduce Phase-Associative Memory (PAM) -- a complex-valued sequence model whose state St ∈ mathbbCd × d accumulates outer products of complex token embeddings retrieved through the conjugate inner product Relangle K mid Qrangle / √d -- and evaluate it against a structurally matched real-valued ablation. Both architectures train stably across a 5M--100M parameter sweep on WikiText-103 under identical conditions; PAM sits at higher absolute loss at every measured scale but improves more rapidly with parameter count, with power-law exponents of -0.15 vs. -0.12 in loss and -0.65 vs. -0.49 in perplexity that narrow the gap between the two architectures monotonically. Further investigation of complex-valued sequence modeling at larger scales could reveal that the loss plateau characteristic of real-valued state-of-the-art language models (e.g. transformers) is reachable with PAM-style architectures with an order of magnitude fewer parameters than the current frontier (~1T), implying that similar capabilities are achievable at sizes runnable on consumer-grade hardware.