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

Ahmed Nebli; Hadi Saadatdoorabi; Kevin Yam

cs.LGFeb 24, 2026

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

Authors: Ahmed Nebli, Hadi Saadatdoorabi, Kevin Yam

Organizations: cAI Technology GmbH

Abstract

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 $Ω(N^2)$ 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.

Figures & tables

Quantum Formalism	Architectural Role	ML Interpretation
Hilbert space $\mathbb{C}^{N}$	Complex latent space with magnitude and phase per coordinate	Hidden state space
Wave function $\ket{\psi(t)}$	Unit-norm complex vector; superposition of interpretations	Recurrent hidden state
Hermitian $H(t){=}H^{\dagger}$	Constrains dynamics to be norm-preserving by construction	Learned transition rule
Schrödinger equation	$H^{\dagger}{=}H\Rightarrow$ unitary $\Rightarrow$ $\\|\psi\\|{\equiv}1$	Recurrence equation
Cayley transform	Structure-preserving discretization; unitarity exact for any step size	Discrete state update
Born rule $\|\!\innerproduct{m_k}{\psi}\!\|^{2}$	Quadratic readout: $N^{2}$ features from $N$ complex dims	Token probabilities

Table 1: Correspondence between quantum-mechanical formalism and the sequence model. Each structure plays a specific functional role: Hermiticity guarantees unitarity, unitarity guarantees norm conservation, and norm conservation ensures the Born rule produces valid probabilities. The architecture uses this algebraic chain as an engineering constraint, not as a physical simulation.

Panel of Figure 1 — Figure 1: Detailed architecture of a single time step in the quantum sequence model. The neural network $g_{\theta}$ (the sole unconstrained learned component, bold outline) receives the token embedding and the current interaction-picture state, and outputs the complex matrix $\Phi(t)$ and real vector $\delta(t)$ . The outer product $\Phi\Phi^{\dagger}$ is Hermitian by construction; adding the learned diagonal $H_{0}$ yields the full Hamiltonian $H(t)$ . The interaction picture removes the known free oscillations, and the Cayley transform discretizes the remaining evolution into an exactly unitary update $W(t)$ . Unitarity preserves $\|\psi\|=1$ at every step, which ensures the Born rule produces a valid probability distribution over the vocabulary. Left-side annotations mark learned components; right-side annotations trace the algebraic guarantees. The dashed line indicates recurrent state feedback, which makes the overall dynamics nonlinear despite each individual step being a linear (unitary) map.

Symbol	Space	Definition
Scalars and index sets
$N$	$\mathbb{Z}_{>0}$	Latent (complex Hilbert-space) dimension
$V$	$\mathbb{Z}_{>0}$	Vocabulary size
$T$	$\mathbb{Z}_{>0}$	Sequence length
$d$	$\mathbb{Z}_{>0}$	Token embedding dimension
$r$	$\mathbb{Z}_{>0}$	Rank of $\Phi(t)$ ; controls the interaction Hamiltonian’s expressivity-efficiency tradeoff ( $r\ll N$ )

Table 2: Summary of all mathematical notation used throughout the paper. All complex vectors are column vectors;

\|\cdot\|

denotes the Euclidean (

L^{2}

) norm;

(\cdot)^{\dagger}

denotes the conjugate transpose.

Panel of Figure 2 — Figure 2: Multi-scale view of the quantum sequence model. (a) The model unrolled over $T$ time steps. At each step the same network $g_{\theta}$ generates a token- and state-dependent Hamiltonian; the Cayley update advances the state on the complex unit sphere; and the Born rule reads out token probabilities. The loss aggregates log-probabilities across all steps. (b) Illustration of the interference mechanism. Processing the disambiguating token “steep” after the prefix “The bank was” causes probability to flow from the financial interpretation to the river interpretation via conserved, antisymmetric probability currents $J_{j\leftarrow k}$ . (c) Detail of the Born-rule readout. The quadratic map $\psi\mapsto\psi\psi^{\dagger}$ lifts the $N$ -dimensional complex state into the $N^{2}$ -dimensional space of Hermitian matrices, exposing both magnitude and phase cross-terms to the linear measurement $\operatorname{tr}(M_{k}\rho)$ .

Panel of Figure 3 — Figure 3: Source of the expressivity separation. Left: The Born-rule readout applies a quadratic (Veronese-type) lifting $\psi\mapsto\psi\psi^{\dagger}$ , promoting the $N$ -dimensional complex state to the $N^{2}$ -dimensional space of rank-one Hermitian matrices. The measurement $\operatorname{tr}(M_{k}\rho)$ is then linear in the lifted space, accessing all $N^{2}$ features including the $\binom{N}{2}$ pairwise phase cross-terms that encode interference. Right: The affine-softmax readout computes $z_{k}=w_{k}^{\top}h+b_{k}$ , which is linear in the $d$ -dimensional real state. The rank constraint (Lemma 5.9 ) limits the log-probability matrix to rank $d{+}2$ . Bottom: Matching the $N^{2}$ features required by the Born-rule target forces $d\geq N^{2}-2$ , a quadratic gap over the complex model’s dimension $N$ , under the full-rank condition on $L^{*}$ stated in Theorem 5.10 .

Sequence Modeling

Sequence modeling studies how to represent, predict, classify, and generate ordered data such as text, time series, events, and biological sequences.

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Phase-Associative Memory: Sequence Modeling in Complex Hilbert Space

Gowrav Vishwakarma, Christopher J. Agostino

Xavoc Technocrats Pvt. Ltd., https: // xavoc. com , India · NPC Worldwide, Bloomington, Indiana 47403, USA

Autoregressive Language Modeling Complex-Valued Neural Networks Associative Memory Quantum-Inspired ML Memory-Augmented Language Models

Open paper AskPDF 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 S_t \in \mathbb{C}^{d \times d} accumulates outer products of complex token embeddings retrieved through the conjugate inner product

\mathrm{Re}\langle K \mid Q\rangle / \sqrt{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 (

\sim

1T), implying that similar capabilities are achievable at sizes runnable on consumer-grade hardware.

Abstract

Sequence Modeling

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