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Sublinear Time Quantum Algorithm for Attention Approximation

ICLR 2026 Conference SubmissionAnonymous Authors
OpenReview Score: 5.5Download Report PDF
quantum computingattention approximationnumerical linear algebra
Abstract:

Given the query, key and value matrices Q,K,V∈Rn×dQ, K, V\in \mathbb{R}^{n\times d}, the attention matrix is defined as Att(Q,K,V)=D−1AV\mathrm{Att}(Q, K, V)=D^{-1}AV where A=exp⁡(QK⊤/d)A=\exp(QK^\top/\sqrt{d}) with exp⁡(⋅)\exp(\cdot) applied entrywise, D=diag(A1n)D=\mathrm{diag}(A{\bf 1}_n). The attention matrix is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix D−1AD^{-1}A incurs Ω(n2)\Omega(n^2), motivating numerous approximation schemes that reduce runtime to O~(nd)\widetilde O(nd) via sparsity or low-rank factorization.

We propose a quantum data structure that approximates any row of Att(Q,K,V)\mathrm{Att}(Q, K, V) using only row queries to Q,K,VQ, K, V. Our algorithm preprocesses these matrices in O~(ϵ−1n0.5(sλ2.5+sλ1.5d+α0.5d))\widetilde{O}\left( \epsilon^{-1} n^{0.5} \left( s_\lambda^{2.5} + s_\lambda^{1.5} d + \alpha^{0.5} d \right) \right) time, where ϵ\epsilon is the target accuracy, sλs_\lambda is the λ\lambda-statistical dimension of the exponential kernel defined by QQ and KK, and α\alpha measures the row distortion of VV that is at most d/srank(V)d/{\rm srank}(V), the stable rank of VV. Each row query can be answered in O~(sλ2+sλd)\widetilde{O}(s_\lambda^2 + s_\lambda d) time.

To our knowledge, this is the first quantum data structure that approximates rows of the attention matrix in sublinear time with respect to nn. Our approach relies on a quantum Nystr{"o}m approximation of the exponential kernel, quantum multivariate mean estimation for computing DD, and quantum leverage score sampling for the multiplication with VV.

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Overview

Overall Novelty Assessment

The paper proposes a quantum data structure for approximating attention matrix rows with sublinear preprocessing and query time, leveraging quantum linear algebra primitives. It resides in the 'Quantum Linear Algebra for Attention' leaf, which contains only four papers total, indicating a relatively sparse research direction within the broader quantum transformer landscape. This leaf sits under 'Quantum Algorithmic Primitives for Attention', distinguishing theory-driven algorithmic approaches from circuit-based or hybrid architectures. The small sibling count suggests this specific intersection of quantum linear algebra and attention approximation remains underexplored compared to more crowded areas like vision transformer hybrids or variational quantum circuits.

The taxonomy reveals neighboring leaves focused on quantum search/optimization primitives and quantum encoding schemes, while sibling branches explore variational circuits and quantum kernel methods. The original paper's emphasis on rigorous complexity bounds and quantum subroutines for matrix operations contrasts with variational approaches that prioritize trainable parameters or kernel methods that leverage Hilbert space embeddings. The taxonomy's scope and exclude notes clarify that works using quantum circuits without explicit linear algebra primitives belong elsewhere, positioning this paper firmly in the algorithmic foundations cluster rather than application-oriented or hybrid architecture branches.

Among six candidates examined across three contributions, no clearly refuting prior work was identified. The core quantum data structure contribution examined one candidate without finding overlap. The Nyström approximation contribution examined zero candidates, leaving its novelty unassessed within this limited search. The row distortion parameter contribution examined five candidates, none providing clear refutation. This limited search scope—six papers from semantic retrieval—means the analysis captures only a narrow slice of potentially relevant quantum algorithms, quantum sampling, or classical attention approximation literature. The absence of refutation among examined candidates suggests novelty within this constrained sample, but does not rule out overlooked prior work in adjacent areas.

Based on top-six semantic matches, the work appears to occupy a sparsely populated niche combining quantum linear algebra with attention approximation guarantees. The limited search scope and small sibling count in the taxonomy suggest either genuine novelty or insufficient coverage of related quantum algorithm literature. The analysis does not assess connections to classical sublinear algorithms for kernel matrices or broader quantum machine learning sampling techniques, which may contain relevant precedents outside the examined candidate set.

Taxonomy

Core-task Taxonomy Papers
50
3
Claimed Contributions
6
Contribution Candidate Papers Compared
0
Refutable Paper

Research Landscape Overview

Core task: quantum algorithm for attention approximation in transformers. The field of quantum transformers has rapidly diversified into several major branches that reflect different research priorities and maturity levels. At the foundational level, Quantum Attention Mechanism Design and Theory explores algorithmic primitives and quantum linear algebra techniques (e.g., Quantum Linear Algebra[10], Fast Quantum Attention[2]) that underpin efficient attention computation, often targeting sublinear or polylogarithmic complexity gains. Hybrid Quantum-Classical Transformer Architectures blend quantum subroutines with classical components to balance near-term hardware constraints and practical performance, as seen in works like Hybrid Quantized Attention[6] and Quantum-Classical Patch Transformers[8]. Domain-Specific Quantum Transformer Applications tailor these mechanisms to specialized tasks such as molecular modeling (Molecular Quantum Transformer[7]), finance (Quantum Attention Finance[15]), and biomedical analysis (Quantum Self-Attention Biomedical[18]), while Fully Quantum Transformer Models and Language Processing pursue end-to-end quantum implementations for natural language tasks (Quantum Self-Attention Text[5], Quantum-Enhanced NLP[16]). Training and implementation branches address the practical challenges of deploying quantum circuits on noisy intermediate-scale quantum (NISQ) devices, and comparative surveys (Survey Quantum Transformers[28]) synthesize progress across these directions. Within the algorithmic primitives cluster, a central tension emerges between theoretical speedup guarantees and practical realizability on current hardware. Sublinear Quantum Attention[0] sits squarely in the Quantum Linear Algebra for Attention subfield, emphasizing rigorous complexity analysis and leveraging quantum subroutines to achieve provable sublinear scaling in sequence length. This contrasts with nearby works like Fast Quantum Attention[2], which may prioritize heuristic approximations or hybrid strategies, and Quantum Adaptive Attention[1], which explores adaptive sampling to balance accuracy and resource costs. Meanwhile, Quantum Transformer Inference[17] focuses on the deployment phase, addressing how to efficiently execute trained models on quantum hardware. The original paper's emphasis on sublinear algorithmic guarantees places it among the more theory-driven efforts, aiming to establish fundamental complexity bounds rather than immediate application readiness, yet it shares the broader goal of making transformer attention tractable in the quantum regime.

Claimed Contributions

Quantum data structure for sublinear-time attention row queries

The authors introduce a quantum data structure that preprocesses query, key, and value matrices using only row queries and achieves sublinear preprocessing time in n. Each row of the attention matrix can then be queried in Õ(s²_λ + s_λd) time, where s_λ is the statistical dimension and α measures row distortion of V.

1 retrieved paper
Quantum Nyström approximation for exponential kernel matrices

The authors develop a quantum algorithm that combines Nyström approximation for the exponential kernel matrix formed from queries and keys, using recursive generalized ridge leverage score sampling implemented via Grover search to achieve sublinear runtime.

0 retrieved papers
Row distortion parameter for leverage score sampling on non-orthonormal matrices

The authors introduce the row distortion parameter α to enable leverage score sampling on value matrices V that do not have orthonormal columns, providing approximate matrix multiplication guarantees in Frobenius norm by sampling Õ(ε⁻²α) rows.

5 retrieved papers

Core Task Comparisons

Comparisons with papers in the same taxonomy category

Contribution Analysis

Detailed comparisons for each claimed contribution

Contribution

Quantum data structure for sublinear-time attention row queries

The authors introduce a quantum data structure that preprocesses query, key, and value matrices using only row queries and achieves sublinear preprocessing time in n. Each row of the attention matrix can then be queried in Õ(s²_λ + s_λd) time, where s_λ is the statistical dimension and α measures row distortion of V.

Contribution

Quantum Nyström approximation for exponential kernel matrices

The authors develop a quantum algorithm that combines Nyström approximation for the exponential kernel matrix formed from queries and keys, using recursive generalized ridge leverage score sampling implemented via Grover search to achieve sublinear runtime.

Contribution

Row distortion parameter for leverage score sampling on non-orthonormal matrices

The authors introduce the row distortion parameter α to enable leverage score sampling on value matrices V that do not have orthonormal columns, providing approximate matrix multiplication guarantees in Frobenius norm by sampling Õ(ε⁻²α) rows.

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