Classical and Quantum Algorithms for Tensor Principal Component Analysis

Matthew B. Hastings

Station Q, Microsoft Research, Santa Barbara, CA 93106-6105, USA
Microsoft Quantum and Microsoft Research, Redmond, WA 98052, USA

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We present classical and quantum algorithms based on spectral methods for a problem in tensor principal component analysis. The quantum algorithm achieves a $quartic$ speedup while using exponentially smaller space than the fastest classical spectral algorithm, and a super-polynomial speedup over classical algorithms that use only polynomial space. The classical algorithms that we present are related to, but slightly different from those presented recently in Ref. [1]. In particular, we have an improved threshold for recovery and the algorithms we present work for both even and odd order tensors. These results suggest that large-scale inference problems are a promising future application for quantum computers.

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Cited by

[1] David Gamarnik, Aukosh Jagannath, and Alexander S. Wein, "Hardness of Random Optimization Problems for Boolean Circuits, Low-Degree Polynomials, and Langevin Dynamics", SIAM Journal on Computing 53 1, 1 (2024).

[2] David Gamarnik, Aukosh Jagannath, and Alexander S. Wein, 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS) 131 (2020) ISBN:978-1-7281-9621-3.

[3] Mohamed Ouerfelli, Mohamed Tamaazousti, and Vincent Rivasseau, "Selective multiple power iteration: from tensor PCA to gradient-based exploration of landscapes", The European Physical Journal Special Topics 232 23-24, 3645 (2023).

[4] Ryan Babbush, Jarrod R. McClean, Michael Newman, Craig Gidney, Sergio Boixo, and Hartmut Neven, "Focus beyond Quadratic Speedups for Error-Corrected Quantum Advantage", PRX Quantum 2 1, 010103 (2021).

[5] Vincent Lahoche, Mohamed Ouerfelli, Dine Ousmane Samary, and Mohamed Tamaazousti, "Field Theoretical Approach for Signal Detection in Nearly Continuous Positive Spectra II: Tensorial Data", Entropy 23 7, 795 (2021).

[6] Alexander M. Dalzell, Sam McArdle, Mario Berta, Przemyslaw Bienias, Chi-Fang Chen, András Gilyén, Connor T. Hann, Michael J. Kastoryano, Emil T. Khabiboulline, Aleksander Kubica, Grant Salton, Samson Wang, and Fernando G. S. L. Brandão, "Quantum algorithms: A survey of applications and end-to-end complexities", arXiv:2310.03011, (2023).

[7] Alexander S. Wein, Ahmed El Alaoui, and Cristopher Moore, "The Kikuchi Hierarchy and Tensor PCA", arXiv:1904.03858, (2019).

[8] Blake A. Wilson, Zhaxylyk A. Kudyshev, Alexander V. Kildishev, Sabre Kais, Vladimir M. Shalaev, and Alexandra Boltasseva, "Machine learning framework for quantum sampling of highly constrained, continuous optimization problems", Applied Physics Reviews 8 4, 041418 (2021).

The above citations are from Crossref's cited-by service (last updated successfully 2024-05-21 07:08:26) and SAO/NASA ADS (last updated successfully 2024-05-21 07:08:26). The list may be incomplete as not all publishers provide suitable and complete citation data.