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So none of those are relevant differences. Moreover, these variants turn out to lead to precisely the same power for quantum computers-namely, the class BQP-as “standard” quantum mechanics, the one over the complex numbers. Quantum mechanics over the reals or the quaternions still has constructive and destructive interference among amplitudes, and unitary transformations, and probabilities that are absolute squares of amplitudes. There are also the real numbers, for starters, and in the other direction, the quaternions. It’s not that complex numbers are a bad choice for the foundation of the deepest known description of the physical universe-far from it! (They’re a field, they’re algebraically closed, they’ve got a norm, how much more could you want?) It’s just that they seem like a specific choice, and not the only possible one. In this post, I’d like to focus on a question that any “explanation” for QM at some point needs to address, in a non-question-begging way: why should amplitudes have been complex numbers? When I was a grad student, it was his relentless focus on that question, and on others in its vicinity, that made me a lifelong fan of Chris Fuchs (see for example his samizdat), despite my philosophical differences with him.
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Short of an ultimate answer, we can at least try to explain why, if you want this or that piece of quantum mechanics, then the rest of the structure is inevitable: why quantum mechanics is an “island in theoryspace,” as I put it in 2003. Why should Nature have been quantum-mechanical? It’s totally unclear what would count as an answer to such a question, and also totally clear that people will never stop asking it.