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. \cite{wein2019kikuchi}. 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.
Highlights
Principal component analysis is a fundamental technique that finds applications in reducing the dimensionality of data and denoising
The runtime bound for the fastest quantum algorithm is in theorem 6. This theorem gives a quartic improvement in the runtime compared to the fastest classical spectral algorithm; more precisely the log of the runtime with the quantum algorithm divided by the log of the runtime of the classical algorithm approaches 1/4 as N → ∞ at fixed N −p/4/λ
We describe the algorithm in this way since, in later classical and quantum algorithms that we give to compute the spectral properties of the matrix, we might not extract the leading eigenvector but instead extract only some vector in this eigenspace due to use of the power method in a classical algorithm or due to approximations in phase estimation in a quantum algorithm
Summary
Principal component analysis is a fundamental technique that finds applications in reducing the dimensionality of data and denoising. In the case p = 4, for example, our Hamiltonian has pairwise interaction terms for all pairs of qudits It is natural from the viewpoint of mean-field theory in physics to expect that the leading eigenvector of the problem, for large nbos, can be approximated by a product state. [1] where it was termed a voting matrix) Implementing this spectral algorithm requires high-dimensional linear algebra, in particular finding the leading eigenvector of a matrix of dimension ≈ N nbos. This makes it a natural candidate for a quantum algorithm. We refer to the latter as a quantum expectation value of O in state ψ to distinguish it from an expectation value over random variables
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