# Classical and Quantum Algorithms for Tensor Principal Component Analysis

## TL;DR

This paper introduces classical and quantum spectral algorithms for tensor principal component analysis, demonstrating quantum speedups and improved classical thresholds, highlighting potential for large-scale inference applications.

## Contribution

It presents novel quantum and classical spectral algorithms for tensor PCA, with quantum algorithms achieving significant speedups and classical algorithms with improved recovery thresholds.

## Key findings

- Quantum algorithm achieves quartic speedup and exponential space reduction.
- Classical algorithms have improved recovery thresholds.
- Results indicate potential for quantum computing in large-scale inference.

## 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.