"So the bottom line is that, for "generic" or "unstructured" search problems, quantum computers can give some speedup over classical computers -- specifically, a quadratic speedup -- but nothing like the exponential speedup of Shor's factoring algorithm."
"the [Church-Turing] thesis has several possible meanings:
1) The universe is equivalent to a Turing machine; thus, computing non-recursive functions is physically impossible. This has been termed the Strong Church–Turing thesis, or Church–Turing–Deutsch principle, and is a foundation of digital physics."
From my knowledge a non-recursive function would be easier than a recursive one, since a non-recursive one will not call itself while a recursive one can. But the quoted phrase seems to imply it must be something more difficult instead...
Problem is, wikipedia's page https://en.wikipedia.org/wiki/Non-recursive_function does not define it either, it just redirects you to the recursion page (though not recursively, so wikipedia did not go for the joke) which does not define it either.
A “non-recursive function” would be more powerful than that; or, conversely, it would be a function which cannot be computed using a Turing machine. Alternatively, these are also know as super-recursive functions [2].
[1] https://en.wikipedia.org/wiki/Computable_function [2] https://en.wikipedia.org/wiki/Super-recursive_algorithm
In other words, the efficiency of every realized model of computation needs to be similar to that of a Turing machine for the thesis to hold. As soon as machines capable of quantum computation are realized, the thesis is broken, as they give superpolynomial speedup over classical Turing machines (assuming things like integer factorization are not in P).
EDIT: I guess it should be noted that there are probably other 'strong Church-Turing theses' hanging around, but I'm fairly certain the folks at Google are referring to this one, which I too am most familiar with. If OP is referring to one which does not require similar efficiency, then the quotation does seem kind of ridiculous. I also agree that the part of the quote that says "current computers cannot replicate" needs to be interpreted with some notion of efficiency as well.