DOCSLIB.ORG

Quantum Algorithms for Classification

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Quantum Algorithms for Classification Quantum algorithms for classification Alessandro Luongo Twitter: @scinawa https://luongo.pro/qml 26-30-2018, Paris, International Conference on Quantum Computing 1 Prologue 2 Toolbox 3 Quantum algorithms for classification.. Quantum Slow Feature Analysis Quantum Frobenius Distance Classifier q-means 4 ..on real data • Classification • Classification Unsupervised methods Supervised methods × × × X 2 Rn d X 2 Rn d; Y 2 Rn m • Anomaly detection • Regression • Clustering • Pattern recognition • Blind signal separation • Time series forecasting • Text mining • Speech recognition Unsupervised methods Supervised methods × × × X 2 Rn d X 2 Rn d; Y 2 Rn m • Anomaly detection • Regression • Clustering • Pattern recognition • Blind signal separation • Time series forecasting • Text mining • Speech recognition • Classification • Classification Unsupervised Unsupervised Unsupervised Supervised We need Quantum Machine Learning! runtime = O(polylog(size)) [HHL09] ... ML’s algos: runtime = O(poly(size)) = O(poly(n; d))... ... but size = O(2time)... ) problem! ML’s algos: runtime = O(poly(size)) = O(poly(n; d))... ... but size = O(2time)... ) problem! We need Quantum Machine Learning! runtime = O(polylog(size)) [HHL09] ... QML team @ IRIF • Iordanis Kerenidis • Jonas Landman • Anupam Prakash Takeaways • There is an efficient quantum procedure for supervised dimensionality reduction: Quantum Slow Feature Analysis • There is an efficient quantum procedure for supervised classification and distance calculation: Quantum Frobenius Distance Estimator . • There is a new efficient quantum procedure for unsupervised classification: q-means • We simulate quantum algorithm on real data: they work! • QRAM based. Xn q 1 k k j i j i P xi i xi n k k2 i=0 xi i=0 • Execution time: O(log nd) • Preparation time: O(nd log nd) • Size: O(nd log nd) 1 - QRAM Let X 2 Rn×d. There is a quantum algorithm that −1 jii j0i ! jii jxii jxii = kxik jxii • Execution time: O(log nd) • Preparation time: O(nd log nd) • Size: O(nd log nd) 1 - QRAM Let X 2 Rn×d. There is a quantum algorithm that −1 jii j0i ! jii jxii jxii = kxik jxii Xn q 1 k k j i j i P xi i xi n k k2 i=0 xi i=0 1 - QRAM Let X 2 Rn×d. There is a quantum algorithm that −1 jii j0i ! jii jxii jxii = kxik jxii Xn q 1 k k j i j i P xi i xi n k k2 i=0 xi i=0 • Execution time: O(log nd) • Preparation time: O(nd log nd) • Size: O(nd log nd) QRAM [[2,3,4],[5,6,7],[8,9,10]] QRAM + swaps... better Thanks Alex Singh for the circuit (ii) jzi such that kjzi − jMxik ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) j + i (iii) a state M≤θM≤θx ~ µ(M)kxk in time O( k + k) δθ M≤θM≤θx Get estimates of kzk = f(M)x (with mult. error ϵ2, time · −1 O() ϵ2 ) Gilyén, András, et al. ”Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.” arXiv preprint arXiv:1806.01838 (2018). 2 - Q-BLAS 2 Rd×d k k - M , s.t. M 2 = 1, in QRAM - x 2 Rd in QRAM. There is a quantum algorithm that w.h.p. returns : −1 (i) jzi such that jzi − jM xi ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) j + i (iii) a state M≤θM≤θx ~ µ(M)kxk in time O( k + k) δθ M≤θM≤θx Get estimates of kzk = f(M)x (with mult. error ϵ2, time · −1 O() ϵ2 ) Gilyén, András, et al. ”Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.” arXiv preprint arXiv:1806.01838 (2018). 2 - Q-BLAS 2 Rd×d k k - M , s.t. M 2 = 1, in QRAM - x 2 Rd in QRAM. There is a quantum algorithm that w.h.p. returns : −1 (i) jzi such that jzi − jM xi ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) (ii) jzi such that kjzi − jMxik ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) Get estimates of kzk = f(M)x (with mult. error ϵ2, time · −1 O() ϵ2 ) Gilyén, András, et al. ”Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.” arXiv preprint arXiv:1806.01838 (2018). 2 - Q-BLAS 2 Rd×d k k - M , s.t. M 2 = 1, in QRAM - x 2 Rd in QRAM. There is a quantum algorithm that w.h.p. returns : −1 (i) jzi such that jzi − jM xi ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) (ii) jzi such that kjzi − jMxik ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) j + i (iii) a state M≤θM≤θx ~ µ(M)kxk in time O( k + k) δθ M≤θM≤θx 2 - Q-BLAS 2 Rd×d k k - M , s.t. M 2 = 1, in QRAM - x 2 Rd in QRAM. There is a quantum algorithm that w.h.p. returns : −1 (i) jzi such that jzi − jM xi ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) (ii) jzi such that kjzi − jMxik ≤ ϵ e in time O(κ(M)µ(M) log(1/ϵ)) j + i (iii) a state M≤θM≤θx ~ µ(M)kxk in time O( k + k) δθ M≤θM≤θx Get estimates of kzk = f(M)x (with mult. error ϵ2, time · −1 O() ϵ2 ) Gilyén, András, et al. ”Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.” arXiv preprint arXiv:1806.01838 (2018). 2.5 - Q-BLAS - A; B 2 Rd×d in QRAM k k k k A 2 = B 2 = 1, in QRAM - x 2 Rd in QRAM. There is a quantum algorithm that w.h.p. returns : −1 (i) jzi such that jzi − j(AB) xi ≤ ϵ (ii) jzi such that kjzi − j(AB)xik ≤ ϵ j + i (iii) a state (AB)≤θ,δ(AB)≤θ,δx Get estimates of kzk = f(AB)x (with mult. error ϵ2, time · −1 O() ϵ2 ) Gilyén, András, et al. ”Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics.” arXiv preprint arXiv:1806.01838 (2018). • Before: Quantum Singular Value Estimation X X αi jvii 7! αi jvii jσii i i • Now: Qubitization: W = eiϕ0σz eiθσx eiϕ1σz eiθσx ··· eiϕkσz eiθσx Family of possible W is large enough... 3 - Compute distances × × V 2 Rn d; C 2 Rk d in the QRAM, and ϵ > 0 ( ) e η There is a quantum algorithm that w.h.p. and in time O ϵ jii jji j0i 7! jii jji jd(vi; cj)i maxkvik where jd(vi; cj) − d(vi; cj)j 6 ϵ , where η = . minkvik Based on: Wiebe, N., Kapoor, A., & Svore, K. (2014). Quantum algorithms for nearest-neighbor methods for supervised and unsupervised learning. arXiv preprint arXiv:1401.2142. 3 - sketch proof • Use Quantum Frobenius Dinstance to build: k k pvi pcj jii jji j0i jvii + jii jji j1i jcji Zij Zij • Hadamard on 3rd qubit. 2 1 2 2 d(vi; cj) p(1)ij = (kvik + cj −2 kvik cj hvi; cji) = 2Zij 2Zij • Perform amplitude estimation on L copies. • Use Median Lemma (Wiebe et. al.) • Invert circuit (garbage collection), multiply by 2Zij. 4 - Tomography For a pure quantum state jxi, there is a tomography algorithm with sample and time complexity O(d log d/ϵ2) that produces an estimate e 2 Rd kek ke − k ≤ x with x 2 = 1 such that x x 2 ϵ with probability at least (1 − 1=d0:83). Kerenidis, Iordanis, and Anupam Prakash. ”A quantum interior point method for LPs and SDPs.” arXiv preprint arXiv:1808.09266 (2018). PCA SFA || FLD AW = BWΛ Slow Feature Analysis (Supervised) Input signal: x(i) 2 Rd. Task: Learn K functions: y(i) = [g1(x(i)); ··· ; gK(x(i))] Such that 8j 2 [K]. Minimize: XK X ( ) 1 2 ∆(yj) = gj(x(s)) − gj(x(t)) a 2 k=1 s;t Tk s<t Constraints on output signal: average of components is 0, variance of components is 1, signals are decorrelated. Def Cov. matrix B := XTX, Derivative cov. matrix A := X_ TX_ Slow Feature Analysis (Supervised) Input signal: x(i) 2 Rd. Task: Learn K functions: y(i) = [g1(x(i)); ··· ; gK(x(i))] Such that 8j 2 [K]. Minimize: XK X ( ) 1 2 ∆(yj) = gj(x(s)) − gj(x(t)) a 2 k=1 s;t Tk s<t Constraints on output signal: average of components is 0, variance of components is 1, signals are decorrelated. Def Cov. matrix B := XTX, Derivative cov. matrix A := X_ TX_ AW = BWΛ Step 1: Whitening Data is whitened (sphered) if B = XTX = I. Whitening its just matrix (Moore-Penrose) inversion Z := X+X .. now ZTZ = I Freebie Theorem! There exists an efficient quantum algorithm for whitening that builds jZi Step 2: Projection • Whiten data jXi 7! jZi • Project data in slow feature space jZi 7! jYi New algo! QSFA P T 2 Rn×d _ 2 Rn log n×d - Let X = i σiuivi , X QRAM. - Let ϵ, θ; δ; η > 0. There exists a quantum algorithm that produces: • j i j j i − j + i j ≤ Y with Y A≤θ,δA≤θ,δZ ϵ in time (( ) ) (µ(X) + µ(X_)) jjZjj ~ κ( )µ( ) log( =") + × O X X 1 jj + jj δθ A≤θ,δA≤θ,δZ • kYk s.t. jkYk − kYk j ≤ η kYk with an additional 1/η factor. New algo! QFDC (Supervised) jTk|×d Xk 2 R matrix of elements labeled k jTk|×d X0 2 R repeats the row x0 for jTkj times.
Load more

Recommended publications
  • Arxiv:1811.00414V3 [Cs.DS] 6 Aug 2021
    Quantum principal component analysis only achieves an exponential speedup because of its state preparation assumptions Ewin Tang∗ University of Washington (Dated: August 10, 2021) A central roadblock to analyzing quantum algorithms on quantum states is the lack of a comparable input model for classical algorithms. Inspired by recent work of the author [E. Tang, STOC’19], we introduce such a model, where we assume we can efficiently perform `2-norm samples of input data, a natural analogue to quantum algorithms that assume efficient state preparation of classical data. Though this model produces less practical algorithms than the (stronger) standard model of classical computation, it captures versions of many of the features and nuances of quantum linear algebra algorithms. With this model, we describe classical analogues to Lloyd, Mohseni, and Rebentrost’s quantum algorithms for principal component analysis [Nat. Phys. 10, 631 (2014)] and nearest-centroid clustering [arXiv:1307.0411]. Since they are only polynomially slower, these algorithms suggest that the exponential speedups of their quantum counterparts are simply an artifact of state preparation assumptions. INTRODUCTION algorithms. In our previous work we suggest an idea for develop- Quantum machine learning (QML) has shown great ing classical analogues to QML algorithms beyond this promise toward yielding new exponential quantum exceptional case [9]: speedups in machine learning, ever since the pioneer- When QML algorithms are compared to clas- ing linear systems algorithm of Harrow, Hassidim, and sical ML algorithms in the context of finding Lloyd [1]. Since machine learning (ML) routines often speedups, any state preparation assumptions push real-world limits of computing power, an exponen- in the QML model should be matched with tial improvement to algorithm speed would allow for ML `2-norm sampling assumptions in the classical systems with vastly greater capabilities.
    [Show full text]
  • Quantum-Inspired Classical Algorithms for Singular Value
    Quantum-Inspired Classical Algorithms for Singular Value Transformation Dhawal Jethwani Indian Institute of Technology (BHU), Varanasi, India [email protected] François Le Gall Nagoya University, Japan [email protected] Sanjay K. Singh Indian Institute of Technology (BHU), Varanasi, India [email protected] Abstract A recent breakthrough by Tang (STOC 2019) showed how to “dequantize” the quantum algorithm for recommendation systems by Kerenidis and Prakash (ITCS 2017). The resulting algorithm, classical but “quantum-inspired”, efficiently computes a low-rank approximation of the users’ preference matrix. Subsequent works have shown how to construct efficient quantum-inspired algorithms for approximating the pseudo-inverse of a low-rank matrix as well, which can be used to (approximately) solve low-rank linear systems of equations. In the present paper, we pursue this line of research and develop quantum-inspired algorithms for a large class of matrix transformations that are defined via the singular value decomposition of the matrix. In particular, we obtain classical algorithms with complexity polynomially related (in most parameters) to the complexity of the best quantum algorithms for singular value transformation recently developed by Chakraborty, Gilyén and Jeffery (ICALP 2019) and Gilyén, Su, Low and Wiebe (STOC 2019). 2012 ACM Subject Classification Theory of computation → Design and analysis of algorithms Keywords and phrases Sampling algorithms, quantum-inspired algorithms, linear algebra Digital Object Identifier 10.4230/LIPIcs.MFCS.2020.53 Related Version A full version of the paper is available at https://arxiv.org/abs/1910.05699. Funding François Le Gall: FLG was partially supported by JSPS KAKENHI grants Nos.
    [Show full text]
  • Taking the Quantum Leap with Machine Learning
    Introduction Quantum Algorithms Current Research Conclusion Taking the Quantum Leap with Machine Learning Zack Barnes University of Washington Bellevue College Mathematics and Physics Colloquium Series [email protected] January 15, 2019 Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion Overview 1 Introduction What is Quantum Computing? What is Machine Learning? Quantum Power in Theory 2 Quantum Algorithms HHL Quantum Recommendation 3 Current Research Quantum Supremacy(?) 4 Conclusion Zack Barnes University of Washington UW Quantum Machine Learning Introduction Quantum Algorithms Current Research Conclusion What is Quantum Computing? \Quantum computing focuses on studying the problem of storing, processing and transferring information encoded in quantum mechanical systems.\ [Ciliberto, Carlo et al., 2018] Unit of quantum information is the qubit, or quantum binary integer. Zack Barnes University of Washington UW Quantum Machine Learning Supervised Uses labeled examples to predict future events Unsupervised Not classified or labeled Introduction Quantum Algorithms Current Research Conclusion What is Machine Learning? \Machine learning is the scientific study of algorithms and statistical models that computer systems use to progressively improve their performance on a specific task.\ (Wikipedia) Zack Barnes University of Washington UW Quantum Machine Learning Uses labeled examples to predict future events Unsupervised Not classified or labeled Introduction Quantum
    [Show full text]
  • Faster Quantum-Inspired Algorithms for Solving Linear Systems
    Faster quantum-inspired algorithms for solving linear systems Changpeng Shao∗1 and Ashley Montanaro†1,2 1School of Mathematics, University of Bristol, UK 2Phasecraft Ltd. UK March 19, 2021 Abstract We establish an improved classical algorithm for solving linear systems in a model analogous to the QRAM that is used by quantum linear solvers. Precisely, for the linear system Ax = b, we show that there is a classical algorithm that outputs a data structure for x allowing sampling and querying to the entries, where x is such that kx−A−1bk ≤ kA−1bk. This output can be viewed as a classical analogue to the output of quantum linear solvers. The complexity of our algorithm 6 2 2 −1 −1 is Oe(κF κ / ), where κF = kAkF kA k and κ = kAkkA k. This improves the previous best 6 6 4 algorithm [Gily´en,Song and Tang, arXiv:2009.07268] of complexity Oe(κF κ / ). Our algorithm is based on the randomized Kaczmarz method, which is a particular case of stochastic gradient descent. We also find that when A is row sparse, this method already returns an approximate 2 solution x in time Oe(κF ), while the best quantum algorithm known returns jxi in time Oe(κF ) when A is stored in the QRAM data structure. As a result, assuming access to QRAM and if A is row sparse, the speedup based on current quantum algorithms is quadratic. 1 Introduction Since the discovery of the Harrow-Hassidim-Lloyd algorithm [16] for solving linear systems, many quantum algorithms for solving machine learning problems were proposed, e.g.
    [Show full text]
  • A Quantum-Inspired Classical Algorithm for Recommendation
    A quantum-inspired classical algorithm for recommendation systems Ewin Tang May 10, 2019 Abstract We give a classical analogue to Kerenidis and Prakash’s quantum recommendation system, previously believed to be one of the strongest candidates for provably expo- nential speedups in quantum machine learning. Our main result is an algorithm that, given an m n matrix in a data structure supporting certain ℓ2-norm sampling op- × erations, outputs an ℓ2-norm sample from a rank-k approximation of that matrix in time O(poly(k) log(mn)), only polynomially slower than the quantum algorithm. As a consequence, Kerenidis and Prakash’s algorithm does not in fact give an exponential speedup over classical algorithms. Further, under strong input assumptions, the clas- sical recommendation system resulting from our algorithm produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n. The main insight of this work is the use of simple routines to manipulate ℓ2-norm sampling distributions, which play the role of quantum superpositions in the classical setting. This correspondence indicates a potentially fruitful framework for formally comparing quantum machine learning algorithms to classical machine learning algo- rithms. arXiv:1807.04271v3 [cs.IR] 9 May 2019 Contents 1 Introduction 2 1.1 QuantumMachineLearning . 2 1.2 RecommendationSystems . 4 1.3 AlgorithmSketch ................................. 6 1.4 FurtherQuestions................................ 7 2 Definitions 8 2.1 Low-RankApproximations. .. 8 2.2 Sampling...................................... 9 3 Data Structure 9 1 4 Main Algorithm 11 4.1 VectorSampling.................................. 11 4.2 Finding a Low-Rank Approximation . ... 13 4.3 ProofofTheorem1................................ 17 5 Application to Recommendations 19 5.1 PreferenceMatrix...............................
    [Show full text]
  • An Overview of Quantum-Inspired Classical Sampling
    An overview of quantum-inspired classical sampling Ewin Tang This is an adaptation of a talk I gave at Microsoft Research in November 2018. I exposit the `2 sampling techniques used in my recommendation systems work and its follow-ups in dequantized machine learning: • Tang -- A quantum-inspired algorithm for recommendation systems • Tang -- Quantum-inspired classical algorithms for principal component analysis and supervised clustering; • Gilyén, Lloyd, Tang -- Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension; • Chia, Lin, Wang -- Quantum-inspired sublinear classical algorithms for solving low-rank linear systems. The core ideas used are super simple. This goal of this blog post is to break down these ideas into intuition relevant for quantum researchers and create more understanding of this machine learning paradigm. Contents An introduction to dequantization . 1 Motivation . 1 The model . 2 Quantum for the quantum-less . 4 Supervised clustering . 5 Recommendation systems . 5 Low-rank matrix inversion . 6 Implications . 7 For quantum computing . 7 For classical computing . 9 Appendix: More details . 9 1. Estimating inner products . 10 2. Thin matrix-vector product with rejection sampling . 10 3. Low-rank approximation, briefly . 11 Glossary . 12 1 Notation is defined in the Glossary. The intended audience is researchers comfortable with probability and linear algebra (SVD, in particular). Basic quantum knowledge helps with intuition, but is not essential: everything from The model onward is purely classical. The appendix is optional and explains the dequantized techniques in more detail. An introduction to dequantization Motivation The best, most sought-after quantum algorithms are those that take in raw, classical input and give some classical output.
    [Show full text]
  • Potential Topic List (Pdf)
    QIC710/CS768/CO681/PHYS767/AM871 Potential Project Topics Please feel free to pursue a project not on this list Quantum machine learning: This is somewhat of a controversial topic. There is extreme interest in using quantum computers for machine learning applications, especially from industry. Very recently, some preprints appeared showing that the performance of some proposed quantum algorithms for machine learning is essentially matched by classical algorithms. Scott Aaronson, “Read the fine print,” https://www.scottaaronson.com/papers/qml.pdf Harrow, Aram W., Avinatan Hassidim, and Seth Lloyd. "Quantum algorithm for linear systems of equations." https://arxiv.org/abs/0811.3171 Seth Lloyd, Silvano Garnerone, Paolo Zanardi, “Quantum algorithms for topological and geometric analysis of big data”, https://arxiv.org/abs/1408.3106 Ewin Tang, “A quantum-inspired classical algorithm for recommendation systems,” https://arxiv.org/abs/1807.04271 Ewin Tang, “Quantum-inspired classical algorithms for principal component analysis and supervised clustering” https://arxiv.org/abs/1811.00414 András Gilyén, Seth Lloyd, Ewin Tang, “Quantum-inspired low-rank stochastic regression with logarithmic dependence on the dimension,” https://arxiv.org/abs/1811.04909 Quantum walks: These can be used as quantum analogues of random walks, and have been shown to be useful for algorithmic purposes. Two survey papers: J. Kempe, “Quantum random walks – an introductory overview”. http://arxiv.org/abs/quant-ph/0303081 A. Ambainis, “Quantum walks and their algorithmic applications”. http://arxiv.org/abs/quant-ph/0403120 Also, a paper by: F. Magniez, A. Nayak, J. Roland, and M. Santha, “Search via Quantum Walk”. http://arxiv.org/abs/quant-ph/0608026 Here are a few more papers in this area.
    [Show full text]
  • Sampling-Based Sublinear Low-Rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning
    Sampling-Based Sublinear Low-Rank Matrix Arithmetic Framework for Dequantizing Quantum Machine Learning Nai-Hui Chia András Gilyén Tongyang Li University of Texas at Austin California Institute of Technology University of Maryland Austin, Texas, USA Pasadena, California, USA College Park, Maryland, USA [email protected] [email protected] [email protected] Han-Hsuan Lin Ewin Tang Chunhao Wang University of Texas at Austin University of Washington University of Texas at Austin Austin, Texas, USA Seattle, Washington, USA Austin, Texas, USA [email protected] [email protected] [email protected] ABSTRACT KEYWORDS We present an algorithmic framework for quantum-inspired clas- quantum machine learning, low-rank approximation, sampling, sical algorithms on close-to-low-rank matrices, generalizing the quantum-inspired algorithms, quantum machine learning, dequan- series of results started by Tang’s breakthrough quantum-inspired tization algorithm for recommendation systems [STOC’19]. Motivated by ACM Reference Format: quantum linear algebra algorithms and the quantum singular value Nai-Hui Chia, András Gilyén, Tongyang Li, Han-Hsuan Lin, Ewin Tang, transformation (SVT) framework of Gilyén et al. [STOC’19], we and Chunhao Wang. 2020. Sampling-Based Sublinear Low-Rank Matrix develop classical algorithms for SVT that run in time independent Arithmetic Framework for Dequantizing Quantum Machine Learning. In of input dimension, under suitable quantum-inspired sampling Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Com- assumptions. Our results give compelling evidence that in the cor- puting (STOC ’20), June 22–26, 2020, Chicago, IL, USA. ACM, New York, NY, responding QRAM data structure input model, quantum SVT does USA, 14 pages.
    [Show full text]
  • A Quantum-Inspired Classical Algorithm for Separable Non
    Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence (IJCAI-19) A Quantum-inspired Classical Algorithm for Separable Non-negative Matrix Factorization Zhihuai Chen1;2 , Yinan Li3 , Xiaoming Sun1;2 , Pei Yuan1;2 and Jialin Zhang1;2 1CAS Key Lab of Network Data Science and Technology, Institute of Computing Technology, Chinese Academy of Sciences, 100190, Beijing, China 2University of Chinese Academy of Sciences, 100049, Beijing, China 3Centrum Wiskunde & Informatica and QuSoft, Science Park 123, 1098XG Amsterdam, Netherlands fchenzhihuai, sunxiaoming, yuanpei, [email protected], [email protected] Abstract of A. To solve the Separable Non-Negative Matrix Factoriza- tions (SNMF), it is sufficient to identify the anchors in the Non-negative Matrix Factorization (NMF) asks to input matrices, which can be solved in polynomial time [Arora decompose a (entry-wise) non-negative matrix into et al., 2012a; Arora et al., 2012b; Gillis and Vavasis, 2014; the product of two smaller-sized nonnegative matri- Esser et al., 2012; Elhamifar et al., 2012; Zhou et al., 2013; ces, which has been shown intractable in general. In Zhou et al., 2014]. Separability assumption is favored by vari- order to overcome this issue, separability assump- ous practical applications. For example, in the unmixing task tion is introduced which assumes all data points in hyperspectral imaging, separability implies the existence are in a conical hull. This assumption makes NMF of ‘pure’ pixel [Gillis and Vavasis, 2014]. And in the topic tractable and is widely used in text analysis and im- detection task, it also means some words are associated with age processing, but still impractical for huge-scale unique topic [Hofmann, 2017].
    [Show full text]
  • Exploring Quantum Machine Learning Algorithms
    CS 499 Project Course Report | Exploring Quantum Machine Learning Algorithms Exploring Quantum Machine Learning Algorithms Harshil Jain [email protected] Anirban Dasgupta [email protected] Computer Science and Engineering IIT Gandhinagar Gujarat, India Abstract Machine learning, as a collection of powerful data analytical methods, is widely used in classification, face recognition, nature language processing, etc. However, the efficiency of machine learning algorithms is seriously challenged by big data. Fortunately, it is found that quantum setting for these algorithms can help overcome this problem. The pace of development in quantum computing mirrors the rapid advances made in machine learning and artificial intelligence. It is natural to ask whether quantum technologies could boost learning algorithms: this field of inquiry is called quantum-enhanced machine learning. In this report, I will describe the quantum recommendation system algorithm, followed by the classical version of it and finally I will describe quantum generative adversarial networks (QGAN) and the results of QGAN on the MNIST dataset. Keywords: Quantum ML, Quantum Recommendation Systems, Quantum GAN 1. Introduction Quantum machine learning is at the crossroads of two of the most exciting current areas of research: quantum computing and classical machine learning. It explores the interaction between quantum computing and machine Learning, investigating how results and tech- niques from one field can be used to solve the problems of the other. With an ever-growing amount of data, current machine learning systems are rapidly approaching the limits of classical computational models. In this sense, quantum computational power can offer ad- vantage in such machine learning tasks.
    [Show full text]
  • Quantum and Quantum-Inspired Recommendation System Algorithms
    Quantum and Quantum-inspired Recommendation System Algorithms Siqi Shen Do June Min Department of Electrical Engineering Department of Computer Science University of Michigan University of Michigan Ann Arbor, MI 48105 Ann Arbor, MI 48105 [email protected] [email protected] Abstract—This report is a case study of how development The report is organized as follows: In SectionII, we briefly of quantum algorithms can not only lead to quantum methods introduce relevant literature in quantum computation and rec- that achieve speedup over classical algorithms but also provide ommendation systems other than the ones covered in this insights for improving classical solutions. Specifically, we focus on the case of collaborative recommendation systems by matrix report. Through Section III,IV, andV, we summarize each sampling and describe the process of how the potential quantum paper while highlighting how the quantum algorithm informed speedup was identified and implemented. Then, we look at an improved classical algorithm. an improved classical algorithm that builds on the quantum algorithm and achieves equivalent computational complexity II. RELATED WORKS and introduce a few guidelines a quantum machine learning researchers may employ. Formal models of quantum computation has been pioneered Index Terms—quantum computation, quantum complexity, by Deutsch in his influential work [1]. Fortnow and Aaronson recommendation systems, low-rank approximation also made important contributions about the relationship of BQP and other complexity classes [3][9]. The approach to I. INTRODUCTION recommendation system covered in this project, collaborative In this report, we examine three quantum computation- filtering, follows the definition and assumptions made in the related papers and provide a synopsis of how development seminal work of Drineas et al [4].
    [Show full text]
  • Major Quantum Computing Advance Made Obsolete by Teenager
    Quanta Magazine Major Quantum Computing Advance Made Obsolete by Teenager 18-year-old Ewin Tang has proven that classical computers can solve the “recommendation problem” nearly as fast as quantum computers. The result eliminates one of the best examples of quantum speedup. By Kevin Hartnett A teenager from Texas has taken quantum computing down a notch. In a paper posted online earlier this month, 18-year-old Ewin Tang proved that ordinary computers can solve an important computing problem with performance potentially comparable to that of a quantum computer. In its most practical form, the “recommendation problem” relates to how services like Amazon and Netflix determine which products you might like to try. Computer scientists had considered it to be one of the best examples of a problem that’s exponentially faster to solve on quantum computers — making it an important validation of the power of these futuristic machines. Now Tang has stripped that validation away. “This was one of the most definitive examples of a quantum speedup, and it’s no longer there,” said Tang, who graduated from the University of Texas, Austin, in spring and will begin a Ph.D. at the University of Washington in the fall. In 2014, at age 14 and after skipping the fourth through sixth grades, Tang enrolled at UT Austin and majored in mathematics and computer science. In the spring of 2017 Tang took a class on quantum information taught by Scott Aaronson, a prominent researcher in quantum computing. Aaronson recognized Tang as an unusually talented student and offered himself as adviser on an independent research project.
    [Show full text]