Church-Turing vs Quantum Computing Illusion

A natural system like the weather can be viewed to perform a form of analog computation as it evolves from one time instant to the next when molecules in the air interact with their neighbors. The computational complexity can be viewed to be polynomial in the size of the physical system. This is expressed in the Physical Church-Turing Thesis PCTT:

• Any physical process can be simulated by a Turing machine.  

Here a Turing machine is a model of a digital computer with polynomial computational capacity capable of simulating a physical process of polynomial computational complexity.

According to PCTT there is thus no physical process expressing exponential complexity, which would be beyond the capacity of digital computing. 

Quantum computing is a form of analog computation with promise of exponential capacity capable of meeting the needs of systems of exponential complexity. It is motivated by a view that quantum mechanics carries exponential complexity in the form of a multi-dimensional wave function and so cannot be simulated on a digital computer. 

We meet here a contradiction:  

• An analog quantum computer is realised in a physical process which according to PCTT is limited to polynomial complexity and so does not have exponential capacity.

We see that PCTT says that a quantum computer with exponential capacity cannot be constructed. No wonder that no such quantum computer has been constructed.

If PCTT is correct, it means that the evolution of a quantum system of atoms and molecules as a physical process, does not expresses polynomial complexity and so in principle can be simulated by digital computation with polynomial capacity. The multi-dimensionality of the wave function appearing to demand exponential capacity thus is an illusion. 

RealQM deconstructs the illusion, by offering simulation of systems of atoms and molecules by digital computation (of polynomial complexity).

PS1 Recall that an N-body simulation has a computational complexity between $N$ and $N^2$ depending on interaction between bodies. 

PS2 If macroscopics has polynomial complexity, then so has microscopics as the basis of macroscopics. If microscopics has exponential complexity, then so has macroscopics based on microscopics.