# Deep Sequence Modeling with Quantum Dynamics: Language as a Wave Function

arXiv: https://arxiv.org/abs/2602.22255 · HTML: https://mlsanity.com/paper/2602.22255/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 34 of its 61 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Mamba: Linear-Time Sequence Modeling with Selective State Spaces](https://mlsanity.com/paper/2312.00752/) (2023)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [FlashAttention: Fast and Memory-Efficient Exact Attention with IO-Awareness](https://mlsanity.com/paper/2205.14135/) (2022)
- [Efficiently Modeling Long Sequences with Structured State Spaces](https://mlsanity.com/paper/2111.00396/) (2021)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [Big Bird: Transformers for Longer Sequences](https://mlsanity.com/paper/2007.14062/) (2020)
- [Probing Classifiers: Promises, Shortcomings, and Advances](https://mlsanity.com/paper/2102.12452/) (2021)
- [Simplified State Space Layers for Sequence Modeling](https://mlsanity.com/paper/2208.04933/) (2022)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Supervised learning with quantum enhanced feature spaces](https://mlsanity.com/paper/1804.11326/) (2018)
- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [Neural Controlled Differential Equations for Irregular Time Series](https://mlsanity.com/paper/2005.08926/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Hyena Hierarchy: Towards Larger Convolutional Language Models](https://mlsanity.com/paper/2302.10866/) (2023)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Liquid Time-constant Networks](https://mlsanity.com/paper/2006.04439/) (2020)
- [Breaking the Softmax Bottleneck: A High-Rank RNN Language Model](https://mlsanity.com/paper/1711.03953/) (2017)
- [Cheap Orthogonal Constraints in Neural Networks: A Simple Parametrization of the Orthogonal and Unitary Group](https://mlsanity.com/paper/1901.08428/) (2019)
- [Pymanopt: A Python Toolbox for Optimization on Manifolds using Automatic Differentiation](https://mlsanity.com/paper/1603.03236/) (2016)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Differentiable Learning of Quantum Circuit Born Machine](https://mlsanity.com/paper/1804.04168/) (2018)
- [On orthogonality and learning recurrent networks with long term dependencies](https://mlsanity.com/paper/1702.00071/) (2017)
- [A quantum-inspired classical algorithm for recommendation systems](https://mlsanity.com/paper/1807.04271/) (2018)
- [Closed-form Continuous-time Neural Models](https://mlsanity.com/paper/2106.13898/) (2021)

## Built on by (most cited first, 2 of 2)

- [The Quantum Shortcut: Complex Phase-State Dynamics Reduce the Optimization Steps of Sequence Models](https://mlsanity.com/paper/2608.14691/) (2026)
- [Phase-HDC: Replacing Optimizer History with Gradient Thresholds in Discrete Phase Learning](https://mlsanity.com/paper/2610.10630/) (2026)

## Abstract (arXiv)

We introduce a sequence modeling framework in which the latent state is a complex-valued wave function evolving on a finite-dimensional Hilbert space under a learned, time-dependent Hamiltonian. Unlike standard recurrent architectures that rely on gating mechanisms to suppress competing hypotheses, our framework utilizes quantum interference: the Hamiltonian steers the phases of complex amplitudes so that conflicting interpretations cancel while compatible ones reinforce. The dynamics are strictly unitary, ensuring that the state norm is preserved exactly at every time step via a Cayley (Crank--Nicolson) discretization. Token probabilities are extracted using the Born rule, a quadratic measurement operator that couples magnitudes and relative phases. Our primary theoretical contribution is a separation theorem characterizing the representational advantage of this readout: we define a family of disambiguation tasks that a complex unitary model of dimension N solves exactly, but which requires a state dimension of Omega(N2) for any real-valued orthogonal model equipped with a standard affine-softmax readout. This quadratic gap arises because the Born rule implicitly lifts the N-dimensional state into the space of rank-one Hermitian matrices, accessing pairwise phase correlations that are inaccessible to linear projections. Finally, we derive a continuity equation for the latent probability mass, yielding conserved pairwise currents that serve as a built-in diagnostic for tracing information flow between dimensions.