I'd like to quote from Nielsen & Chuang, Quantum Computation and Quantum Information, 10th anniversary edition, page 5 (emphasis mine):
One class of challenges to the strong Church–Turing thesis comes from the field of analog computation. In the years since Turing, many different teams of researchers have noticed that certain types of analog computers can efficiently solve problems believed to have no efficient solution on a Turing machine. At first glance these analog computers appear to violate the strong form of the Church–Turing thesis. Unfortunately for analog computation, it turns out that when realistic assumptions about the presence of noise in analog computers are made, their power disappears in all known instances; they cannot efficiently solve problems which are not efficiently solvable on a Turing machine. This lesson – that the effects of realistic noise must be taken into account in evaluating the efficiency of a computational model – was one of the great early challenges of quantum computation and quantum information, a challenge successfully met by the development of a theory of quantum error-correcting codes and fault-tolerant quantum computation. Thus, unlike analog computation, quantum computation can in principle tolerate a finite amount of noise and still retain its computational advantages.
Is this a statement that noise scales faster than some power of the problem size, or can someone point me in the right direction so that I can find out more about whether these scaling limits are fundamental or merely an "engineering issue"?
To be clear, I am asking if analog computers cannot beat Turing machines in efficiency due to noise.