# 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

### Abstract

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] 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).

[2] 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).

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

The above citations are from Crossref's cited-by service (last updated successfully 2021-09-23 07:06:13) and SAO/NASA ADS (last updated successfully 2021-09-23 07:06:15). The list may be incomplete as not all publishers provide suitable and complete citation data.