  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.
