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arXiv:2608.14691

The Quantum Shortcut: Complex Phase-State Dynamics Reduce the Optimization Steps of Sequence Models

Ahmed Nebli, Hadi Saadatdoorabi, Christopher Keibel, Kevin Yam

cs.LGquant-phSubmitted Aug 7, 2026• Announced Aug 10, 2026• 0 citations
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Abstract

Sequence models are conventionally distinguished by their backbone, the mechanism that routes information across positions, such as attention or recurrence. This paper varies a choice that is prior to the backbone and shared by nearly all current models: the \emph{substrate}, the number system in which the hidden state is represented together with the form of the map from state to prediction. The prevailing substrate is a real-valued state with an affine--softmax readout; we study a complex-valued alternative drawn from the mathematics of quantum theory, in which information is carried by the phases of the state and scores are quadratic Born forms. Prior work proved an idealized version of this substrate representationally stronger than any real model with a linear readout; we ask whether it also trains faster. Relaxing the two properties that block deployment, exact unitarity and the Born vocabulary readout, we instantiate it in the Mamba state-space model and an attention-based Transformer. At 253M parameters, matched to within $0.02\%$ and trained under one fixed protocol on three byte-level corpora, the complex models reach every measured validation loss in approximately one third (state-space) and one half (attention) of the optimization steps of their real counterparts. The two backbones then diverge. Once the learning-rate warmup ends, the state-space advantage continues to widen, from $0.321$ to $0.354$ bits per character on OpenWebText and from $0.368$ to $0.396$ on FineWeb, which an artifact of the warmup ramp would not do; the attention advantage instead decays toward zero on every corpus, and is therefore an effect of early training.

Related papers

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

Ahmed Nebli, Hadi Saadatdoorabi, Kevin Yam

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.

cs.LGCross-listedarXiv:2602.22255•Feb 26, 2026•0 citations

Born Attention is Linear

Marcel Crâşmaru

Softmax self-attention costs O(T^2) in the sequence length T because the exponential kernel has no finite-dimensional feature map, so linear-time variants approximate it. We observe that Born-rule attention, whose weights are quantum-measurement probabilities a_kj ∝ |<Q_k, K_j>|^2 over complex queries and keys, is an exact finite kernel with the rank-one density-matrix feature map ρ_x = x x†, since |<Q_k, K_j>|^2 = tr(ρ_Qk ρ_Kj). As the complex degree-two case of polynomial-kernel linear attention it admits an exact O(T d^2) causal recurrence and constant-state decoding (d^2 (d_v + 1) reals per head, with d_v the real value width, independent of T), and it keeps rotary positions exactly. For GPT-2's head geometry (64 real channels paired into d = 32 complex) the feature dimension is d^2 = 1024. Measured on an A100, compiled constant-state decoding runs from 6.4 MB per layer at 0.12 ms per token at any length, against a 3.2 GB FlashAttention-2 cache at 2.26 ms per token at 1M tokens; unfused prefill overtakes FlashAttention-2 between 64k and 128k tokens (11x at 1M), and a fused form (Triton feature lift plus a chunkwise linear-attention kernel) between 16k and 32k (4x at 128k), reaching training parity at 64k. The linear and quadratic forms are two implementations of one operator; whether that operator matches softmax is a separate modeling question, answered here only at small scale: the p = 1 operator bolted into a real-valued GPT trails softmax by 0.12 nats at context 1k and 0.77 at 16k (attention dilution; a decay kept exactly in the d^2 state restores the 0.12 gap at 16k, and p = 2 sharpening with decay recovers 74% of the 1k gap), whereas inside a fully complex residual stream it matches softmax at 1k and keeps a quarter of the real-stream gap at 4k; gated linear attention beats both there. At 128k context on PG-19 the variable that matters is forgetting: plain Born dilutes, Born with a fixed decay carried in its state is 0.32 nats ahead of softmax and within 0.04 of gated linear attention, which stays 1.8x faster. Streamed with their constant state over the PG-19 test partition in 5M-byte windows, two 4M-complex-parameter byte models of the source framework score 1.66 (linear Born with decay) and 1.70 (gated) bits per byte, where a 350M-parameter hierarchical Mamba–Transformer stack built for that window reports 2.45. Preprint, October 2026. Also submitted to arXiv (identifier to be added on announcement). Code: the CNet framework, https://github.com/crasmarum/CNet .

•Oct 27, 2026•0 citations

QLAM: A Quantum Long-Attention Memory Approach to Long-Sequence Token Modeling

Hoang-Quan Nguyen, S. D. Pandey, Ky Luu

Modeling long-range dependencies in sequential data remains a central challenge in machine learning. Transformers address this challenge through attention mechanisms, but their quadratic complexity with respect to sequence length limits scalability to long contexts. State-space models (SSMs) provide an efficient alternative with linear-time computation by evolving a latent state through recurrent updates, but their memory is typically formed via additive or linear transitions, which can limit their ability to capture complex global interactions across tokens. In this work, we introduce one of the first studies to leverage the superposition property of quantum systems to enhance state-based sequence modeling. In particular, we propose Quantum Long-Attention Memory (QLAM), a hybrid quantum-classical memory mechanism that can be viewed as a quantum extension of state-space models. Instead of maintaining a classical latent state updated through additive dynamics, QLAM represents the hidden state as a quantum state whose amplitudes encode a superposition of historical information. The state evolves through parameterized quantum circuits conditioned on the input, enabling a non-classical, globally update mechanism. In this way, QLAM preserves the recurrent and linear-time structure of SSMs while fundamentally enriching the memory representation through quantum superposition. Unlike attention mechanisms that explicitly compute pairwise interactions, QLAM implicitly captures global dependencies through the evolution of the quantum state, and retrieves task-relevant information via query-dependent measurements. We evaluate QLAM on sequential variants of standard image classification benchmarks, including sMNIST, sFashion-MNIST, and sCIFAR-10, where images are flattened into token sequences. Across all tasks, QLAM consistently improves over recurrent baselines and transformer-based models.

arXiv:2605.13833•May 13, 2026•0 citations

A Quantum Roadmap for Softmax Attention: Exact Born-Rule Analogs for Softmax Attention on the Probability Simplex

Eric A. F. Reinhardt, Adam J. Hauser

The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of $π$/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is $ε$-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.

quant-pharXiv:2608.11173•Aug 12, 2026•0 citations

Multi-Stage Mamba-Based Architecture for Fast and Scalable Superconducting Qubit Readout

Luca Otting, Xiaorang Guo, Emmanouil Giortamis +3 more

Reliable qubit readout is a critical bottleneck toward fault-tolerant quantum computing (FTQC). In superconducting quantum processors, readout operations are both error-prone and high-latency. These challenges become more severe in frequency-multiplexed architectures, where signal crosstalk among neighboring qubits significantly degrades readout fidelity. Existing machine learning (ML)-based approaches rely on feed-forward neural networks (FNNs) that suffer from large parameter sizes and lack an end-to-end network that jointly addresses relaxation errors and discriminates qubit states. In this work, we present a multi-stage qubit state discriminator based on the Mamba model, which enables efficient sequence modeling with linear complexity. The first stage performs initial state discrimination, followed by a refinement stage that identifies and mitigates relaxation-induced errors. Our lightweight model achieves a geometric mean readout fidelity of 0.906, outperforming the best-reported state-of-the-art method while reducing parameter size by 49.6%; our optimal model further reaches 0.911. Both models remain robust across varying input trace lengths, maintaining a high fidelity of 0.893 at readout durations as short as 500 $ns$, achieving up to a 26% reduction in logical error rate over prior work in quantum error correction (QEC).

quant-pharXiv:2607.11668•Jul 14, 2026•0 citations

Trainable Quantum-Native Language Models: Dissipative Memory, Capacity Limits, and Hardware Validation

Deniz Yunus Göğüş

We study byte-level autoregressive language models whose hidden state is a small quantum register, whose transitions are learned token-conditioned quantum channels, and whose next-byte probabilities come directly from Born-rule measurements without a learned classical vocabulary projection or softmax. In the fixed-width recurrent setting, unitary dynamics cannot contract or reset stored distinctions; experimentally, unitary recurrence loses to its retrained one-token control in every tested variant. Introducing learned dissipation, then token-conditioned dissipative gating, yields useful memory beyond one token. The gated model improves test negative log-likelihood by 0.20 +/- 0.03 nats/byte over retrained parameter-matched one-token controls at context length 12, and by 0.29-0.33 at longer contexts, but never beats retrained two-token controls. A key-value recall benchmark shows a width-dependent threshold: one stored pair meets the pre-registered accuracy bar at 4-5 memory qubits, two pairs at 6-7, and three pairs remain unsolved. This is consistent with, but does not establish, a capacity explanation for the short effective memory horizon. As a secondary result, one-qubit models reach lower stationary memory entropy than a heuristically optimized classical predictive frontier at matched error in 9/9 seeds. On IBM Heron hardware, computational-basis memory-register distributions show signal against a uniform null in 12/12 circuits across prefixes of up to 12 tokens, next-bit probabilities pass pre-registered accuracy bars at four-token prefixes, and bytes are generated autoregressively by sampling physically measured bit probabilities. The tested models are classically simulable and substantially worse than classical n-gram baselines. We claim no computational quantum advantage.

quant-pharXiv:2610.12806•Oct 12, 2026•0 citations
Sequence models are conventionally distinguished by their backbone, the mechanism that routes information across positions, such as attention or recurrence. This paper varies a choice that is prior to the backbone and shared by nearly all current models: the \emph{substrate}, the number system in which the hidden state is represented together with the form of the map from state to prediction. The prevailing substrate is a real-valued state with an affine--softmax readout; we study a complex-valued alternative drawn from the mathematics of quantum theory, in which information is carried by the phases of the state and scores are quadratic Born forms. Prior work proved an idealized version of this substrate representationally stronger than any real model with a linear readout; we ask whether it also trains faster. Relaxing the two properties that block deployment, exact unitarity and the Born vocabulary readout, we instantiate it in the Mamba state-space model and an attention-based Transformer. At 253M parameters, matched to within 0.02%0.02\%0.02% and trained under one fixed protocol on three byte-level corpora, the complex models reach every measured validation loss in approximately one third (state-space) and one half (attention) of the optimization steps of their real counterparts. The two backbones then diverge. Once the learning-rate warmup ends, the state-space advantage continues to widen, from 0.3210.3210.321 to 0.3540.3540.354 bits per character on OpenWebText and from 0.3680.3680.368 to 0.3960.3960.396 on FineWeb, which an artifact of the warmup ramp would not do; the attention advantage instead decays toward zero on every corpus, and is therefore an effect of early training.
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 NNN solves exactly, but which requires a state dimension of Ω(N2)Ω(N^2)Ω(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 NNN-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.
Softmax self-attention costs O(T^2) in the sequence length T because the exponential kernel has no finite-dimensional feature map, so linear-time variants approximate it. We observe that Born-rule attention, whose weights are quantum-measurement probabilities a_kj ∝ |<Q_k, K_j>|^2 over complex queries and keys, is an exact finite kernel with the rank-one density-matrix feature map ρ_x = x x†, since |<Q_k, K_j>|^2 = tr(ρ_Qk ρ_Kj). As the complex degree-two case of polynomial-kernel linear attention it admits an exact O(T d^2) causal recurrence and constant-state decoding (d^2 (d_v + 1) reals per head, with d_v the real value width, independent of T), and it keeps rotary positions exactly. For GPT-2's head geometry (64 real channels paired into d = 32 complex) the feature dimension is d^2 = 1024. Measured on an A100, compiled constant-state decoding runs from 6.4 MB per layer at 0.12 ms per token at any length, against a 3.2 GB FlashAttention-2 cache at 2.26 ms per token at 1M tokens; unfused prefill overtakes FlashAttention-2 between 64k and 128k tokens (11x at 1M), and a fused form (Triton feature lift plus a chunkwise linear-attention kernel) between 16k and 32k (4x at 128k), reaching training parity at 64k. The linear and quadratic forms are two implementations of one operator; whether that operator matches softmax is a separate modeling question, answered here only at small scale: the p = 1 operator bolted into a real-valued GPT trails softmax by 0.12 nats at context 1k and 0.77 at 16k (attention dilution; a decay kept exactly in the d^2 state restores the 0.12 gap at 16k, and p = 2 sharpening with decay recovers 74% of the 1k gap), whereas inside a fully complex residual stream it matches softmax at 1k and keeps a quarter of the real-stream gap at 4k; gated linear attention beats both there. At 128k context on PG-19 the variable that matters is forgetting: plain Born dilutes, Born with a fixed decay carried in its state is 0.32 nats ahead of softmax and within 0.04 of gated linear attention, which stays 1.8x faster. Streamed with their constant state over the PG-19 test partition in 5M-byte windows, two 4M-complex-parameter byte models of the source framework score 1.66 (linear Born with decay) and 1.70 (gated) bits per byte, where a 350M-parameter hierarchical Mamba–Transformer stack built for that window reports 2.45. Preprint, October 2026. Also submitted to arXiv (identifier to be added on announcement). Code: the CNet framework, https://github.com/crasmarum/CNet .
Modeling long-range dependencies in sequential data remains a central challenge in machine learning. Transformers address this challenge through attention mechanisms, but their quadratic complexity with respect to sequence length limits scalability to long contexts. State-space models (SSMs) provide an efficient alternative with linear-time computation by evolving a latent state through recurrent updates, but their memory is typically formed via additive or linear transitions, which can limit their ability to capture complex global interactions across tokens. In this work, we introduce one of the first studies to leverage the superposition property of quantum systems to enhance state-based sequence modeling. In particular, we propose Quantum Long-Attention Memory (QLAM), a hybrid quantum-classical memory mechanism that can be viewed as a quantum extension of state-space models. Instead of maintaining a classical latent state updated through additive dynamics, QLAM represents the hidden state as a quantum state whose amplitudes encode a superposition of historical information. The state evolves through parameterized quantum circuits conditioned on the input, enabling a non-classical, globally update mechanism. In this way, QLAM preserves the recurrent and linear-time structure of SSMs while fundamentally enriching the memory representation through quantum superposition. Unlike attention mechanisms that explicitly compute pairwise interactions, QLAM implicitly captures global dependencies through the evolution of the quantum state, and retrieves task-relevant information via query-dependent measurements. We evaluate QLAM on sequential variants of standard image classification benchmarks, including sMNIST, sFashion-MNIST, and sCIFAR-10, where images are flattened into token sequences. Across all tasks, QLAM consistently improves over recurrent baselines and transformer-based models.
The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of πππ/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is εεε-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.
Reliable qubit readout is a critical bottleneck toward fault-tolerant quantum computing (FTQC). In superconducting quantum processors, readout operations are both error-prone and high-latency. These challenges become more severe in frequency-multiplexed architectures, where signal crosstalk among neighboring qubits significantly degrades readout fidelity. Existing machine learning (ML)-based approaches rely on feed-forward neural networks (FNNs) that suffer from large parameter sizes and lack an end-to-end network that jointly addresses relaxation errors and discriminates qubit states. In this work, we present a multi-stage qubit state discriminator based on the Mamba model, which enables efficient sequence modeling with linear complexity. The first stage performs initial state discrimination, followed by a refinement stage that identifies and mitigates relaxation-induced errors. Our lightweight model achieves a geometric mean readout fidelity of 0.906, outperforming the best-reported state-of-the-art method while reducing parameter size by 49.6%; our optimal model further reaches 0.911. Both models remain robust across varying input trace lengths, maintaining a high fidelity of 0.893 at readout durations as short as 500 nsnsns, achieving up to a 26% reduction in logical error rate over prior work in quantum error correction (QEC).
We study byte-level autoregressive language models whose hidden state is a small quantum register, whose transitions are learned token-conditioned quantum channels, and whose next-byte probabilities come directly from Born-rule measurements without a learned classical vocabulary projection or softmax. In the fixed-width recurrent setting, unitary dynamics cannot contract or reset stored distinctions; experimentally, unitary recurrence loses to its retrained one-token control in every tested variant. Introducing learned dissipation, then token-conditioned dissipative gating, yields useful memory beyond one token. The gated model improves test negative log-likelihood by 0.20 +/- 0.03 nats/byte over retrained parameter-matched one-token controls at context length 12, and by 0.29-0.33 at longer contexts, but never beats retrained two-token controls. A key-value recall benchmark shows a width-dependent threshold: one stored pair meets the pre-registered accuracy bar at 4-5 memory qubits, two pairs at 6-7, and three pairs remain unsolved. This is consistent with, but does not establish, a capacity explanation for the short effective memory horizon. As a secondary result, one-qubit models reach lower stationary memory entropy than a heuristically optimized classical predictive frontier at matched error in 9/9 seeds. On IBM Heron hardware, computational-basis memory-register distributions show signal against a uniform null in 12/12 circuits across prefixes of up to 12 tokens, next-bit probabilities pass pre-registered accuracy bars at four-token prefixes, and bytes are generated autoregressively by sampling physically measured bit probabilities. The tested models are classically simulable and substantially worse than classical n-gram baselines. We claim no computational quantum advantage.