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Diffusion Language Models as Open Quantum Systems

Akinori Tanaka, Yukinao Akamatsu, Takuki Tomita, Yuji Hirono

arXiv:2610.21672v1cs.LGcs.CLquant-ph

TL;DR

The paper asks how continuous-time diffusion-language-model dynamics can be embedded in open quantum systems to extend diffusion sampling. It develops sampling for superposed sentence states and confidence-based remasking; experiments find retained influence from unselected conditions and highest mean judge preference with an intermediate remasking onset.

  • Problem

    The paper asks how continuous-time diffusion-language-model dynamics can be represented in open quantum systems as a basis for extending diffusion sampling.

    Source: §1 · p. 1

  • Method

    The authors embed diffusion dynamics in Lindblad dynamics, use τ-leaping unraveling to generate from Superposition of Prompts and Thoughts, and derive confidence-based remasking through perturbations of masked diffusion.

    Source: §1 · pp. 1–2, §3.3 · p. 7

  • Results

    Superposition outputs retain information from unselected conditioning components, while intermediate-onset remasking achieves the highest mean judge preference across tested checkpoints.

    Source: §5 · p. 9

  • Takeaways & Limitations

    In these experiments, lower perplexity and higher grammatical acceptability do not necessarily translate into better overall text quality, so evaluation should extend beyond perplexity.

    Source: §4.2 · p. 9, §5 · p. 9

    The empirical evaluation covers only the pretrained models and generation settings studied; effectiveness at larger scales and in conditional generation remains unestablished.

    Source: H LIMITATIONS · p. 48

Abstract

from arXiv · show

Diffusion language models generate text by reversing a classical diffusion process over token sequences. We show that their continuous-time forward and reverse dynamics can be embedded in the framework of open quantum systems, which describes quantum systems interacting with their environment. This connection provides a basis for extending diffusion sampling to superpositions of sentence states and for modifying the underlying generation dynamics. We construct a $\tau$-leaping unraveling scheme that simulates stochastic trajectories of these states while updating multiple token positions at each step. This enables generation conditioned on superposed prompts or thoughts, which we call Superposition of Prompts (SoP) and Superposition of Thoughts (SoT). The same formulation yields Quantum-Inspired Remasking (QIR): perturbing the masked-diffusion dynamics produces a confidence-based rule for revising previously generated tokens. Experiments with pretrained diffusion language models show that superposition-based generation can retain information from conditioning components not selected at the final measurement, while QIR improves LLM-assessed text quality when applied at an appropriate stage of generation.

1 INTRODUCTION

The paper embeds continuous-time diffusion language models in open quantum-system dynamics, then uses that framework to introduce superposed sentence generation and quantum-inspired remasking. It positions this formulation against prior work on diffusion language models, remasking, and quantum connections to diffusion.

1 INTRODUCTION

  • Sentence states remain distinct alternatives in a superposition, with their components and relative weights evolving together rather than as independent samples.The paper contrasts superposition with concatenation.Source: §1 · pp. 1–2
  • Stochastic unraveling with τ-leaping updates multiple token positions per step, supporting Superposition of Prompts and Superposition of Thoughts; a final measurement selects an output sentence.The same formulation perturbs masked-diffusion dynamics to derive confidence-based Quantum-Inspired Remasking.Source: §1 · p. 2
  • The framework embeds continuous-time discrete diffusion language models in Lindblad dynamics, recovering reverse diffusion in the diagonal sector and allowing modified dynamics or superposed sentence states.The paper distinguishes its exact embedding of discrete token dynamics from prior connections based on a semiclassical approximation.Source: §1 · pp. 1–2, §1.1 · p. 3
  • Experiments report that superposition outputs retain information from unselected conditioning components, while appropriately timed QIR improves judge-assessed quality; applying QIR too early increases repetition despite lowering perplexity.These findings distinguish text quality from perplexity alone.Source: §1 · p. 2

1.1 RELATED WORKS

  • Prior work covers uniform and masked diffusion and several remasking strategies; QIR instead derives stochastic remasking rates from perturbed Lindblad dynamics.The related-work discussion also places this paper among quantum generative models and physics-based analyses of diffusion.Source: §1.1 · pp. 2–3

2 BACKGROUND

The background presents discrete diffusion as a forward corruption process whose generation procedure reverses that process, and introduces open-system dynamics as a framework that contains classical distributions as a special case. A diagonal embedding connects the two, while the Lindblad formulation also supplies reverse-time dynamics.

2.1 DISCRETE DIFFUSION MODELS

  • Discrete diffusion gradually corrupts token sequences in a forward process, and generation proceeds by reversing that process.The sequence distribution evolves according to a master equation and its reverse-time counterpart.Source: §2.1 · p. 3
  • A common transition changes one token position, allowing the model to estimate single-token scores instead of probability ratios between arbitrary sequence pairs.The transition rate is time-dependent, and the score estimates the concrete probability ratios needed for reverse sampling.Source: §2.1 · p. 3

2.2 OPEN QUANTUM SYSTEMS AND THE LINDBLAD EQUATION

  • A token-sequence distribution can be embedded as a diagonal density operator, making discrete diffusion open-system dynamics restricted to the diagonal sector.Going beyond that sector allows coherent superpositions of sequences.Source: §2.2 · pp. 3–4
  • The Lindblad equation governs Markovian open-system dynamics, with Hamiltonian terms generating coherent evolution and jump terms describing environment-induced dissipative transitions.It preserves positivity and unit trace of the density operator.Source: §2.2 · p. 4
  • Reverse-time Lindblad operators can be defined from the forward dynamics; for diagonal embeddings, the reverse-time equation reduces to discrete diffusion's reverse-time dynamics.The paper uses this connection to formulate inference as stochastic unraveling of reverse open-system dynamics.Source: §2.2 · p. 4

3 A QUANTUM FORMULATION OF DIFFUSION LANGUAGE MODELS

The paper embeds forward and reverse diffusion language-model dynamics in the Lindblad framework, then uses stochastic unraveling to extend sampling to superposed sentence states and perturbed dynamics. It introduces SoP, SoT, and QIR as applications of that formulation.

3.1 EMBEDDING DIFFUSION LANGUAGE MODELS INTO OPEN QUANTUM SYSTEMS

  • The Lindblad equation reproduces classical forward diffusion and, under time reversal, its reverse diffusion dynamics.Source: §3.1 · pp. 4–5
  • Masked diffusion treats the mask as an absorbing lower-energy state; adding reverse jumps with γ perturbations motivates QIR.Source: §3.1 · p. 5

3.2 SIMULATIONS: STOCHASTIC UNRAVELING

  • τ-leaping unraveling updates multiple token positions per step and reproduces reverse-time Lindblad dynamics up to first order in ∆t.Source: §3.2 · pp. 5–6
  • An ensemble of stochastic pure-state trajectories reproduces density-operator dynamics, while projective measurement yields a discrete sentence.Source: §3.2 · p. 6, Figure 2 · p. 6

3.3 QUANTUM-INSPIRED SAMPLINGS

  • SoP applies stochastic unraveling to sentence-state superpositions conditioned on different prompts, then measures one output sentence.Source: §3.3 · p. 6
  • SoT replaces SoP's prompt segment with superposed reasoning tokens; mixed-thought outputs are expected when the model supports reasoning mode.Source: §3.3 · p. 6
  • QIR estimates token reconstruction confidence from the pretrained model and remasks tokens with lower confidence at higher rates.Source: §3.3 · p. 7

4 EXPERIMENTS

The experiments test whether superposed prompts and thoughts produce blended text, and how QIR timing affects generation quality. Superposition retains signal from unselected conditions, while middle-onset QIR earns the highest mean judge preference among tested schedules.

4.1 EXPERIMENT 1: SENTENCE GENERATION WITH TEXT SUPERPOSITION

  • The forest–snow experiments use descriptive prompts for SoP and prescriptive prompts for SoT to test semantic blending, not to outperform conditional-generation methods.A regularization parameter β controls mixture degree because component weights can concentrate rapidly and make superposition disappear early in denoising.Source: §4.1 · p. 7
  • Uniform diffusion concentrates superposition weights onto one component faster than masked diffusion in these experiments, suggesting less stable superpositions.The experiments disable remasking for the forest–snow comparison.Source: §4.1 · p. 7
  • SoP outputs occupy an intermediate forest–snow similarity region and show greater similarity to the unselected condition than single-prompt outputs.The representative SoT sample fuses the divergent stylistic constraints of the forest and snow prompts into one narrative.Source: §4.1 · p. 7, Figure 3 · p. 8

4.2 EXPERIMENT 2: QUANTUM-INSPIRED REMASKING

  • 62.0% for SEDD-medium, 62.4% for RADD-medium, and 61.8% for iLLaDA-8B-Base were the Qwen preference scores with middle-onset QIR against no-remasking baselines.This schedule had the highest mean preference among evaluated schedules; earlier remasking lowered perplexity and raised CoLA scores but reduced preference and token diversity while increasing repetition.Source: Table 2 · p. 9, §4.2 · p. 9

5 CONCLUSION AND OUTLOOK

The paper embeds continuous-time diffusion language models in Lindblad dynamics and uses the framework for superposed conditioning and quantum-inspired remasking. Forest–snow experiments show that unselected conditioning can remain reflected after measurement, while intermediate-onset remasking achieves the highest mean judge preference among tested checkpoints.

5 CONCLUSION AND OUTLOOK

  • Superposition outputs show higher similarity to the unselected forest-or-snow condition than single-condition baselines, suggesting its influence persists after final measurement.QIR uses perturbed jump operators; intermediate onset achieves the highest mean judge preference across tested checkpoints, while earlier onset lowers perplexity but can increase repetition.Source: §5 · p. 9
  • The framework embeds continuous-time diffusion language models into Lindblad dynamics, enabling sampling from superposed conditioning states.Source: §5 · p. 9
  • Future work includes reducing remasking's computational cost and extending its evaluation to conditional generation.Source: §5 · p. 9

AI USE STATEMENT

This section discloses the authors’ use of generative AI and reviews the diffusion-language-model and open-quantum-system concepts that frame the paper’s methods. It develops the open-system dynamics and stochastic quantum-jump simulation approach, and identifies where proofs of the main theorems are provided.

AI USE STATEMENT

  • Generative AI assisted with brainstorming, manuscript readability and organization, and software implementation, but authors verified suggestions and retain responsibility for the work.The tools were not used autonomously to produce scientific claims, proofs, or result interpretations.Source: AI USE STATEMENT · p. 10

A NOTATIONS

  • The notation defines token sequences over a vocabulary, uses token ID 0 for the mask, and distinguishes forward-time from backward-time quantities.It also introduces operators and sequence-level Hilbert spaces for the quantum formulation.Source: A NOTATIONS · p. 15

B.1 DIFFUSION PROCESS

  • A valid reverse-time diffusion generator reverses transition directions and weights rates by probability ratios; simply negating the forward generator generally yields invalid negative off-diagonal rates.The stated reverse-process result assumes positive state probabilities over the time interval considered.Source: §B.1 · pp. 16–17

B.2.1 SAMPLING 1-TOKEN PER STEP

  • Directly evolving the full sequence distribution is intractable, so standard trajectory sampling uses first-order transition probabilities and changes at most one token per step.Because the generator allows only single-token transitions, sequence generation can require many sequential steps.Source: §B.2.1 · p. 18, §B.2.2 · p. 18

B.2.2 FORWARD SOLUTION

  • The forward transition kernel factorizes across token positions, enabling independent token updates in one step and motivating approximate reverse-time τ-leaping.The reverse generator’s time-dependent probability ratios prevent direct application of the forward solution, so its treatment is approximate.Source: §B.2.2 · p. 19, §B.2.3 · p. 19

B.3 TRANSFORMATION OF THE n0-PREDICTOR INTO A SCORE ESTIMATOR

  • The review relates clean-state n0 prediction to score estimation by matching the model’s reverse conditional probabilities to the diffusion-based reverse transition.It describes this relation for uniform and masked diffusion formulations.Source: §B.3 · pp. 19–21

C.2 QUANTUM STATE AND MEASUREMENT

  • Quantum states are represented by density operators, which include pure and mixed states and determine measurement probabilities through the Born rule.The language-model formulation uses a sequence Hilbert space built from token-state spaces.Source: §C.2 · pp. 22–23

C.3 DYNAMICS OF OPEN QUANTUM SYSTEMS

  • Open quantum systems interact with an environment, and their dynamics use completely positive trace-preserving maps that carry valid quantum states to valid states.These maps are quantum analogues of Markov kernels; complete positivity also preserves positivity when an auxiliary system is included.Source: §C.3 · p. 23
  • The Lindblad framework generalizes classical diffusion dynamics to quantum states and admits a time-reversed formulation analogous to diffusion-model reverse dynamics (Kwon & Kim, 2018).The reverse formulation is described in terms of time-backward operators.Source: §C.3 · p. 24

C.4.1 UNRAVELING 1-JUMP PER STEP

  • The quantum-jump method simulates the Lindblad equation with a stochastic differential equation driven by jump processes and defines an effective Hamiltonian to simplify notation.Source: §C.4.1 · p. 24

D PROOFS OF THE MAIN THEOREMS

D.1 PROOF OF THEOREM 3.1

This section proves Theorem 3.1 by relating classical diffusion to diagonal Lindblad dynamics and developing results on time reversal, sampling, and a τ-leaping unraveling for text-basis and superposition states. It also derives the unraveling’s operator-action norm and uses it to normalize the resulting vector.

D.1 PROOF OF THEOREM 3.1

  • Theorem 3.1 follows by reducing the Lindblad equation to the classical diffusion equation when the density operator is diagonal.The proof identifies p_t(m) with the diagonal entries of ρ_t and sets off-diagonal entries to zero.Source: §D.1 · pp. 26, 28

D.2 PROOF OF THEOREM 3.2

  • Theorem 3.2 follows by showing the time-reversal Hamiltonian vanishes and the time-reversal jump operators meet the conditions for Lindblad-to-diffusion reduction.The proof applies Theorem D.1 to the time-reversal Lindblad equation and derives the explicit time-reversal jump operators.Source: §D.2 · p. 29

D.3 SAMPLING BY STOCHASTIC UNRAVELING AND MEASUREMENT

  • Stochastic unraveling followed by projective measurement samples from the conditional probability p_t|s(m|n) of the diffusion process.The proof establishes that the resulting measurement distribution and the conditional diffusion probability satisfy the same ODE and initial condition.Source: §D.3 · p. 30

D.4 EXPLICIT FORMS OF τ-LEAPING QUANTUM STATES

  • The simulation setup specifies a required format for token-level jump operators, while the following derivation gives token-level time-reversal Hamiltonian and jump-operator forms.The text identifies the token-level Hamiltonian and jump operator in the derivation.Source: §D.4 · p. 31, §D.4.1 · pp. 31–32

D.4.2 τ-LEAPING UNRAVELING

  • τ-leaping addresses the inefficiency of exact unravelings, which permit only one token jump per step, by enabling updates at multiple token positions.The proposed method uses token-position-decomposed jumps and matches conventional diffusion-language-model τ-leaping for text-basis states.Source: §D.4.2 · p. 32, §D.4.3 · p. 36
  • The τ-leaping procedure is extended from text-basis states to superpositions by sampling token-wise and determining the updated state for each sentence component.For distinct sentence basis states, only the selected token position jumps in each component.Source: §D.4.3 · pp. 35–37
  • The derivation calculates the operator-action norm under the condition that distinct indices correspond to distinct token positions, then divides the vector by the square root of that value.The authors say this normalization completes the proof of the claim.Source: §D.4.3 · p. 38

E IMPLEMENTATION DETAILS

The implementation stores weighted sentence components and applies shared stochastic token transitions, with multiple positions able to update in one step. SoP and SoT condition generation on paired inputs, while QIR adds timed remasking based on reciprocal model scores.

E IMPLEMENTATION DETAILS

  • The implementation stores token IDs in a (B, M, N) array and component weights in a (B, M) array, avoiding vocabulary-sized one-hot vectors at every position.Vocabulary dimensions appear when evaluating model scores and transition rates.Source: E IMPLEMENTATION DETAILS · p. 38
  • Different positions can jump in the same discrete update, with a shared sampled transition applied across components and incompatible components receiving zero unregularized jump weight.A no-jump event leaves component sequences unchanged.Source: E IMPLEMENTATION DETAILS · p. 39

E.2 SUPERPOSITION OF PROMPTS (SOP)

  • SoP conditions each component on a fixed prefix and optional suffix, restores these tokens during sampling, and uses a configured sequence length that includes fixed tokens.Different prefix lengths can therefore yield different numbers of generated tokens at the same configured length.Source: §E.2 · p. 39

E.3 SUPERPOSITION OF THOUGHTS (SOT)

  • SoT holds the supplied thought fixed and generates a continuation after its closing tag; tag placement specifies input format, not evidence that the model reasons.The fixed prompt pairs used in the experiments are listed in Table 4.Source: §E.3 · p. 40, Table 4 · p. 40

E.4 QUANTUM-INSPIRED REMASKING (QIR)

  • QIR estimates remasking by masking an unmasked token, evaluating the model's diffusion score for restoring that token, and taking the score's inverse.The inverse applies to the diffusion score, not a raw logit.Source: §E.4 · pp. 40–41
  • QIR can be restricted to the start (t ≤ 0.9), middle (t ≤ 0.5), or end (t ≤ 0.1) of denoising; computation can sample K unmasked tokens uniformly.The remasking update is skipped when no positions are eligible; finite-step saturation can turn excessive remasking mass into a certain remask.Source: §E.4 · p. 41

F SUPPLEMENTARY SUPERPOSITION EXPERIMENTS

Supplementary material documents the forest/snow experiment and its similarity table, alongside further prompt and component-weight analyses. It also notes a possible speedup using shadow tokens and attention masks.

F SUPPLEMENTARY SUPERPOSITION EXPERIMENTS

  • The appendix supplements the forest/snow results with conditioning prompts, a similarity-evaluation protocol, thought-superposition experiments, and component-weight dynamics under masked and uniform diffusion.The weight-dynamics analysis examines the coefficient-update exponent β.Source: F SUPPLEMENTARY SUPERPOSITION… · p. 41
  • A shadow-token approximation with appropriate attention masks is proposed as a possible way to speed inference.The passage attributes this approach to Hong et al. (2026).Source: F SUPPLEMENTARY SUPERPOSITION… · p. 41
  • Table 5 reports average cosine similarities between generated-text groups and single-prefix reference conditions, with sample counts and uncertainties.Its rows distinguish generated-text groups and its columns identify the single-prefix reference; bold values mark similarity to the condition not selected at final measurement.Source: Table 5 · p. 42

F.1 SOP EXPERIMENT DETAILS

In the forest/snow experiment, superposition outputs were more similar to the reference condition not selected at final measurement than single-condition outputs. Similarity to the selected condition decreased, so cosine similarity alone does not establish that every output combines both conditions.

F.1 SOP EXPERIMENT DETAILS

  • 0.4477 versus 0.3898: forest-selected superposition outputs had higher similarity to the snow reference than forest-only generation.For snow-selected outputs, similarity to the forest reference rose from 0.3898 to 0.4294.Source: §F.1 · p. 42
  • 0.5053 versus 0.5367: forest-selected superposition outputs had lower similarity to the forest reference than forest-only generation.For snow-selected outputs, similarity to the snow reference fell from 0.5439 to 0.5252.Source: §F.1 · p. 42
  • Cosine similarity alone does not establish that every output combines the two conditions, and marginal uncertainties alone do not test the differences.The reported shifts are consistent with retaining information associated with the unselected component.Source: §F.1 · p. 42

F.2 SOT EXPERIMENT DETAILS

The appendix gives a generated SoT example and examines how component weights evolve under masked and uniform diffusion. Weight persistence depends on the prompt pair and update exponent, and the checkpoint comparison is limited by different pretrained weights.

F.2 SOT EXPERIMENT DETAILS

  • Table 1 shows a (chopped) generated sentence from pretrained GSAI-ML/iLLaDA-8B-Instruct for entries i = 1 and i = 2 of story_reasoning__forest_snow in Table 4, with equal initial weights, remasking disabled, and β = 0.01.Source: §F.2 · p. 42

F.3 WEIGHT DYNAMICS IN MASKED AND UNIFORM DIFFUSION

  • The component-weight experiment uses masked SEDD-small and uniform diffusion with two fixed prompt pairs, equal initial weights, 256 steps, and 120 trajectories per setting.It compares β = 0.01 and 0.1 while changing only the coefficient-update exponent; remasking is disabled.Source: §F.3 · p. 43, Figure 4 · p. 43
  • At β = 0.1, final past/future entropy is 0.355 ± 0.024 bits for masked SEDD and 0.180 ± 0.026 bits for uniform SEDD, with uniform SEDD declining earlier.Forest/city remains near balance: 0.981 ± 0.002 bits for masked SEDD and 0.975 ± 0.004 bits for uniform SEDD; at β = 0.01, both prompt pairs remain close to one bit.Source: §F.3 · p. 43, Figure 5 · p. 44
  • Increasing β preserves more imbalance in coefficient updates, allowing preference for one component to accumulate more strongly, but does not explain the checkpoint difference by itself.The forest/city results show little concentration even at β = 0.1, so the effect is prompt-dependent.Source: §F.3 · pp. 43–44

G SUPPRIMENTARY REMASKING EXPERIMENT

This appendix details unconditional remasking comparisons against sampling without remasking, including generation settings, text-quality measures, pairwise judging, and supplementary repetition statistics. The supplied material specifies procedures and figure encodings but does not state the plotted outcomes.

G SUPPRIMENTARY REMASKING EXPERIMENT

  • The appendix documents the protocol, automatic metrics, pairwise evaluation, and supplementary repetition statistics for the quantum-inspired remasking comparison.The comparison examines how remasking onset relates to generated-text quality relative to sampling without remasking.Source: G SUPPRIMENTARY REMASKING EXPERIMENT · p. 44

G.1 GENERATION PROTOCOL

  • Remasking comparisons use the stochastic-unraveling decoder on unconditional generation from a fully masked sequence; none disables remasking rather than switching to Euler decoding.The onset indices for end, middle, and start are 102, 512, and 921 in the 1024-step run; K = 8 samples candidate positions, and γ scales the remasking contribution linearly.Source: §G.1 · p. 44

G.2 AUTOMATIC METRICS

  • Evaluation reports unigram token-frequency entropy in bits, textattack/roberta-base-CoLA classifier probabilities for LABEL_1 on truncated inputs, and repeated token n-gram fractions.The CoLA classifier truncates inputs at its limit. Repetition is analyzed for n = 2–8 on 1024-token sequences using 29 windows at stride 32; Figure 7 shows n = 7, 8.Source: §G.2 · p. 45

G.3 OPEN-MODEL PAIRWISE EVALUATION

  • Qwen3-32B compares each remasked passage with its same-model, same-onset, same-seed no-remasking passage in both A/B orders; invalid pairs are excluded without re-evaluation.The judge assesses readability, fluency, coherence, progression, and unnecessary repetition, and may choose a tie.Source: §G.3 · pp. 45–46, §G.3.1 · p. 46

G.3.1 SYSTEM PROMPT

  • The shared judge prompt treats passages as data, asks for overall readable-text quality rather than factual verification, and requires A, B, or tie with a concrete reason.The generated passages are supplied separately as JSON fields A and B.Source: §G.3.1 · p. 46

G.4 STATISTICAL SUMMARIES

  • Figure 6 plots four text-quality metrics by generation checkpoint, while Figure 7 plots same-seed repetition differences from no remasking in percentage points.Intervals are sequence-level bootstrap intervals; repetition bars resample paired sequence differences, not windows, and the statistical summaries report no multiple-comparison correction or direct onset-difference significance test.Source: §G.4 · p. 46, Figure 6 · p. 47, Figure 7 · p. 47

H LIMITATIONS

The paper identifies added model evaluations and limited empirical scope as constraints on the remasking results.

H LIMITATIONS

  • Remasking requires extra model evaluations, and its effectiveness beyond the studied pretrained models and generation settings, including in conditional generation, remains unestablished.The paper also does not establish quantum speedup or provide an efficient quantum circuit implementation.Source: H LIMITATIONS · p. 48

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