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arXiv:2509.16801

Sublinear Time Quantum Sensitivity Sampling

Zhao Song, David Paul Woodruff, Lichen Zhang

Published Sep 20, 2025• 0 citations
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Abstract

We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems. Our unified framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation. Our contributions include: * $k$-median and $k$-means clustering: For $n$ points in $d$-dimensional Euclidean space, we give an algorithm that constructs an $ε$-coreset in time $\widetilde O(n^{0.5}dk^{2.5}~\mathrm{poly}(ε^{-1}))$ for $k$-median and $k$-means clustering. Our approach achieves a better dependence on $d$ and constructs smaller coresets that only consist of points in the dataset, compared to recent results of [Xue, Chen, Li and Jiang, ICML'23]. * $\ell_p$ regression: For $\ell_p$ regression problems, we construct an $ε$-coreset of size $\widetilde O_p(d^{\max\{1, p/2\}}ε^{-2})$ in time $\widetilde O_p(n^{0.5}d^{\max\{0.5, p/4\}+1}(ε^{-3}+d^{0.5}))$, improving upon the prior best quantum sampling approach of [Apers and Gribling, QIP'24] for all $p\in (0, 2)\cup (2, 22]$, including the widely studied least absolute deviation regression ($\ell_1$ regression). * Low-rank approximation with Frobenius norm error: We introduce the first quantum sublinear-time algorithm for low-rank approximation that does not rely on data-dependent parameters, and runs in $\widetilde O(nd^{0.5}k^{0.5}ε^{-1})$ time. Additionally, we present quantum sublinear algorithms for kernel low-rank approximation and tensor low-rank approximation, broadening the range of achievable sublinear time algorithms in randomized numerical linear algebra.

DOI
10.48550/arxiv.2509.16801

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We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems. Our unified framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation. Our contributions include: * kkk-median and kkk-means clustering: For nnn points in ddd-dimensional Euclidean space, we give an algorithm that constructs an εεε-coreset in time O~(n0.5dk2.5 poly(ε−1))\widetilde O(n^{0.5}dk^{2.5}~\mathrm{poly}(ε^{-1}))O(n0.5dk2.5 poly(ε−1)) for kkk-median and kkk-means clustering. Our approach achieves a better dependence on ddd and constructs smaller coresets that only consist of points in the dataset, compared to recent results of [Xue, Chen, Li and Jiang, ICML'23]. * ℓp\ell_pℓp​ regression: For ℓp\ell_pℓp​ regression problems, we construct an εεε-coreset of size O~p(dmax⁡{1,p/2}ε−2)\widetilde O_p(d^{\max\{1, p/2\}}ε^{-2})Op​(dmax{1,p/2}ε−2) in time O~p(n0.5dmax⁡{0.5,p/4}+1(ε−3+d0.5))\widetilde O_p(n^{0.5}d^{\max\{0.5, p/4\}+1}(ε^{-3}+d^{0.5}))Op​(n0.5dmax{0.5,p/4}+1(ε−3+d0.5)), improving upon the prior best quantum sampling approach of [Apers and Gribling, QIP'24] for all p∈(0,2)∪(2,22]p\in (0, 2)\cup (2, 22]p∈(0,2)∪(2,22], including the widely studied least absolute deviation regression (ℓ1\ell_1ℓ1​ regression). * Low-rank approximation with Frobenius norm error: We introduce the first quantum sublinear-time algorithm for low-rank approximation that does not rely on data-dependent parameters, and runs in O~(nd0.5k0.5ε−1)\widetilde O(nd^{0.5}k^{0.5}ε^{-1})O(nd0.5k0.5ε−1) time. Additionally, we present quantum sublinear algorithms for kernel low-rank approximation and tensor low-rank approximation, broadening the range of achievable sublinear time algorithms in randomized numerical linear algebra.
kkk-Clustering in Rd\mathbb{R}^dRd (e.g., kkk-median and kkk-means) is a fundamental machine learning problem. While near-linear time approximation algorithms were known in the classical setting for a dataset with cardinality nnn, it remains open to find sublinear-time quantum algorithms. We give quantum algorithms that find coresets for kkk-clustering in Rd\mathbb{R}^dRd with O~(nkd3/2)\tilde{O}(\sqrt{nk}d^{3/2})O~(nk​d3/2) query complexity. Our coreset reduces the input size from nnn to poly(kε−1d)\mathrm{poly}(kε^{-1}d)poly(kε−1d), so that existing ααα-approximation algorithms for clustering can run on top of it and yield (1+ε)α(1 + ε)α(1+ε)α-approximation. This eventually yields a quadratic speedup for various kkk-clustering approximation algorithms. We complement our algorithm with a nearly matching lower bound, that any quantum algorithm must make Ω(nk)Ω(\sqrt{nk})Ω(nk​) queries in order to achieve even O(1)O(1)O(1)-approximation for kkk-clustering.
Regression is a cornerstone of statistics and machine learning, with applications spanning science, engineering, and economics. While quantum algorithms for regression have attracted considerable attention, most existing work has focused on linear regression, leaving many more complex yet practically important variants unexplored. In this work, we present a unified quantum framework for accelerating a broad class of regression tasks -- including linear and multiple regression, Lasso, Ridge, Huber, ℓp\ell_pℓp​-, and δpδ_pδp​-type regressions -- achieving up to a quadratic improvement in the number of samples mmm over the best classical algorithms. This speedup is achieved by extending the recent classical breakthrough of Jambulapati et al. (STOC'24) using several quantum techniques, including quantum leverage score approximation (Apers &Gribling, 2024) and the preparation of many copies of a quantum state (Hamoudi, 2022). For problems of dimension nnn, sparsity r<nr < nr<n, and error parameter εεε, our algorithm solves the problem in O~(rmn/ε+poly(n,1/ε))\widetilde{O}(r\sqrt{mn}/ε+ \mathrm{poly}(n,1/ε))O(rmn​/ε+poly(n,1/ε)) quantum time, demonstrating both the applicability and the efficiency of quantum computing in accelerating regression tasks.
Given the query, key and value matrices Q,K,V∈Rn×dQ, K, V\in \mathbb{R}^{n\times d}Q,K,V∈Rn×d, the attention module is defined as Att(Q,K,V)=D−1AV\mathrm{Att}(Q, K, V)=D^{-1}AVAtt(Q,K,V)=D−1AV where A=exp⁡(QK⊤/d)A=\exp(QK^\top/\sqrt{d})A=exp(QK⊤/d​) with exp⁡(⋅)\exp(\cdot)exp(⋅) applied entrywise, D=diag(A1n)D=\mathrm{diag}(A{\bf 1}_n)D=diag(A1n​). The attention module is the backbone of modern transformers and large language models, but explicitly forming the softmax matrix D−1AD^{-1}AD−1A incurs Ω(n2)Ω(n^2)Ω(n2) time, motivating numerous approximation schemes that reduce runtime to O~(nd)\widetilde O(nd)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)Att(Q,K,V) using only row queries to Q,K,VQ, K, VQ,K,V. Our algorithm preprocesses these matrices in O~(ε−1n0.5(sλ2.5+sλ1.5d+α0.5d))\widetilde{O}\left( ε^{-1} n^{0.5} \left( s_λ^{2.5} + s_λ^{1.5} d + α^{0.5} d \right) \right)O(ε−1n0.5(sλ2.5​+sλ1.5​d+α0.5d)) time, where εεε is the target accuracy, sλs_λsλ​ is the λλλ-statistical dimension of the exponential kernel defined by QQQ and KKK, and ααα measures the row distortion of VVV that is at most d/srank(V)d/{\rm srank}(V)d/srank(V), the stable rank of VVV. Each row query can be answered in O~(sλ2+sλd)\widetilde{O}(s_λ^2 + s_λd)O(sλ2​+sλ​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 nnn. Our approach relies on a quantum Nyström approximation of the exponential kernel, quantum multivariate mean estimation for computing DDD, and quantum leverage score sampling for the multiplication with VVV.
We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffice to generalize all prior results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ 2 -norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.
We present an algorithmic framework for quantum-inspired classical algorithms on close-to-low-rank matrices, generalizing the series of results started by Tang’s breakthrough quantum-inspired algorithm for recommendation systems [STOC’19]. Motivated by quantum linear algebra algorithms and the quantum singular value transformation (SVT) framework of Gilyén et al. [STOC’19], we develop classical algorithms for SVT that run in time independent of input dimension, under suitable quantum-inspired sampling assumptions. Our results give compelling evidence that in the corresponding QRAM data structure input model, quantum SVT does not yield exponential quantum speedups. Since the quantum SVT framework generalizes essentially all known techniques for quantum linear algebra, our results, combined with sampling lemmas from previous work, suffices to generalize all recent results about dequantizing quantum machine learning algorithms. In particular, our classical SVT framework recovers and often improves the dequantization results on recommendation systems, principal component analysis, supervised clustering, support vector machines, low-rank regression, and semidefinite program solving. We also give additional dequantization results on low-rank Hamiltonian simulation and discriminant analysis. Our improvements come from identifying the key feature of the quantum-inspired input model that is at the core of all prior quantum-inspired results: ℓ2-norm sampling can approximate matrix products in time independent of their dimension. We reduce all our main results to this fact, making our exposition concise, self-contained, and intuitive.
We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved O~(n+m)\widetilde{O}(\sqrt{n}+\sqrt{m})O(n​+m​) dependence on matrix dimension nnn and constraint number mmm. Compared with the O~(mn)\widetilde{O}(mn)O(mn) runtime of existing classical methods, this suggests a quartic quantum speedup when m≈nm \approx nm≈n. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let sss denote the the input matrix sparsity and γ:=Rr/εγ:=Rr/\varepsilonγ:=Rr/ε capture dependence on the primal (R)(R)(R) and dual (r)(r)(r) radii as well as target accuracy (ε)(\varepsilon)(ε). When γ2≤min⁡{m,n/s}γ^2\leq\min\{m,n/s\}γ2≤min{m,n/s}, our solver runs in time O~(nsγ4.5+msγ2)\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)O(nsγ4.5+msγ2). For γ=O(1)γ=O(1)γ=O(1), this is O~((n+m)s)\widetilde{O}\left((n+m)s\right)O((n+m)s) and sublinear in the O(mns)O(mns)O(mns) input size. Similar to the quantum algorithms, this matches known lower bounds with respect to mmm and nnn, up to logarithmic factors. This implies that, with respect to dimensions mmm and nnn, there is no super-quadratic quantum advantage for generic sparse SDP solving.