A quantum-inspired classical algorithm for recommendation systems
Ewin Tang
- 23 Jun 2019
- pp 217-228
TL;DR: A classical analogue to Kerenidis and Prakash’s quantum recommendation system is given, previously believed to be one of the strongest candidates for provably exponential speedups in quantum machine learning, which produces recommendations exponentially faster than previous classical systems, which run in time linear in m and n.
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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 exponential 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 operations, 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 classical 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 algorithms.
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References
Matrix Factorization Techniques for Recommender Systems
TL;DR: As the Netflix Prize competition has demonstrated, matrix factorization models are superior to classic nearest neighbor techniques for producing product recommendations, allowing the incorporation of additional information such as implicit feedback, temporal effects, and confidence levels.
Quantum Computing in the NISQ era and beyond
TL;DR: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future as mentioned in this paper, which will be useful tools for exploring many-body quantum physics, and may have other useful applications.
Quantum algorithm for linear systems of equations.
TL;DR: This work exhibits a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa, and proves that any classical algorithm for this problem generically requires exponentially more time than this quantum algorithm.
Quantum Computing in the NISQ era and beyond
John Preskill
- 06 Aug 2018
TL;DR: Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future, and the 100-qubit quantum computer will not change the world right away - but it should be regarded as a significant step toward the more powerful quantum technologies of the future.
Strengths and Weaknesses of Quantum Computing
TL;DR: It is proved that relative to an oracle chosen uniformly at random with probability 1 the class $\NP$ cannot be solved on a quantum Turing machine (QTM) in time $o(2^{n/2})$.
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