This article is full of false statements:
Here are some
(1)"a quantum computer is exponentially more efficient at performing Fourier Transforms than a classical computer."
That is totally false. A QC can calculate faster than classical computers only "quantum" Fourier transforms, not "classical" Fourier transforms. They are quite different. For a classical Fourier transform, you can "print" all N components of the answer at once, for a quantum Fourier transform, you can print ONLY ONE of the N components of the answer. The other N-1 components are lost when you print just one of them.
(2)"Today, the problem is approached by sampling, using the Metropolis algorithm on a classical computer. The European/Canadian group propose an alteration to that algorithm that uses a quantum computer to obtain the samples."
Again, very misleading.
The statement refers to the following paper:
Quantum Metropolis Sampling
K. Temme, T.J. Osborne, K.G. Vollbrecht, D. Poulin, F. Verstraete
The problem with this paper is that it is not very honest. It doesn't tell
you that their QC algorithm is no faster than classical!. There are QC algorithms, which PREDATE the Temme et al al algorithm and which can sample faster than the Temme et al algorithm. Those algorithms use Szegedy operators. (See, for example, the work of Pawel Wocjan). The Temme et al paper doesn't compare their algorithm with those because they know they would look quite bad in the comparison.
(3)"Another example is here: a quantum algorithm for solving linear equations (where you have a matrix and a known vector, and wish to compute an unknown vector)."
That is totally false. Again, with the classical algorithm for solving LE, you can print all N components of the answer, but with the quantum one you can print ONLY ONE component. Equating these two algorithms is quite disingenuous. Check out this post by Eric Dexler, one of the most prominent figures of nanotech