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Quoting from this blog post by Earl T. Campbell:
Magic states are a special ingredient, or resource, that allows quantum computers to run faster than traditional computers.
One interesting example that is mentioned in that blog post is that, in the case of a single qubit, any state apart from the eigenstates of the Pauli matrices is magic.
How are these magic states more generally defined? Is it really just any state that is not a stabilizer state, or is it something else?
It is any state that, if you have an unlimited supply of them, can be used to give you universal quantum computation when used in conjunction with perfect Clifford operations.
The standard example is that if you can produce the state $(|0\rangle+e^{i\pi/4}|1\rangle)/\sqrt{2}$, then you can combine this with Clifford operations in order to apply a $T$ gate (see Fig. 10.25 in Nielsen and Chuang), and we know that $T$+Clifford is universal.
To be clear, in the one qubit case that is being discussed, I assume the accurate statement is that any pure state that is not an eigenstate of a Pauli operator is magic.
The real interest is in mixed states - how noisy can a particular magic state be before it isn’t magic any more. The theory being that Clifford operations are often comparatively easy in a fault-tolerant scenario (they can be applied transversally), and it is creating the one non-Clifford gate that’s hard. The more noise it can tolerate, the easier it will be to make.
I believe that I’ve seen results proving that there are some non-Clifford mixed states which are not magic, but I don’t remember the reference off the top of my head. Earl’s papers are the ones you want to read on this topic.
