[tezthenerd]

Let me present it a different way.

Someone comes to you with two formal models of computing. Both models involve representing the state of the computer as a vector of real numbers, they both involve finite dimensional subsystems combined with a tensor product, both involve gates defined over the reals also combined via the tensor product and so on. That is, both models are just about evolution of a vector in some (very high) dimensional real vector space according to gates acting on a small number of subsystems. In fact these two models are identical, except for the fact that in model A the readout procedure involves computing a property of the output vector with the 1-norm, while in model B the 2-norm is used.

This is not an analogy, these are valid mathematical formulations of classical and quantum computing, the correspondences (and differences!) are well understood and rigorous.

Now you read an IEEE article that vociferously objects to the feasibility of building a computer based on model B, but all the objections are to do with properties of model B that it fully shares with model A. And model A you know can be very well approximated already in the physical world, which means reality was somehow was not inhibited by those objections. To try and refute the physicality of model B with an argument based on premises already satisfied by model A is silly.

(Note that even if it were the case that complex numbers were necessary for quantum computing, which they are not - see eg. my book Q is for Quantum - you can map the quantum density matrix on n qubits to a real vector over the basis of Hermitian matrices).

[dhammack]

Hey I'm not familiar with this way of comparing classical and quantum computation. Can you point me to some more details? I have Nielson's book but don't remember seeing this analogy before!

[tezthenerd]

I presume its explained in Scott Asronsons book, its implicitly there in Nielsen and Chuang. But the best way to understand it is by example - try to write out how you would describe classical probabilistic computation on two classical bits to mimic the quantum circuit type of picture, and if you succeed the generalization will be obvious.

[xpuente]

> ... as a vector of real numbers...

Using IEEE754?