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8

To my mind, this theorem is not very well stated in this form, if taken out of context. Where it says "phase gates", this may be misleading. It means specifically just $S=\sqrt{Z}$ and not what I think of as a phase gate, which can have an arbitrary phase (but they have very specifically introduced their terminology about 3 pages earlier). This is a key ...

3

Another way to think about this: To simulate what goes on in a quantum computer we have to do a lot of matrix math using $(2^N \times 2^N)$ matrices$^1$, and the action of (most) of the clifford gates can be actually be accomplished by applying some non-linear, low complexity matrix operation instead of a matrix multiplication. For example, the Pauli-X gate,...

3

Is this just bad phrasing (or a typo) on Wikipedia's side, or am I missing something? It does sound like bad phrasing. The idea here is that our set of observables is not necessarily mutually (pairwise) commuting. If you have three observables $A$, $B$ and $C$, and $[A,B]=0$ and $[B,C]=0$, then you're right that the measurement of $A$ will have no effect on ...

2

You're presumably thinking of a spectrum with classical mechanics at one end and quantum mechanics at another, with some hazy "classical-quantum" in between. That's not a great way to think about it. Classical mechanics is more of a practical approximation of quantum mechanics under certain conditions (cf. classical limit), as per the correspondence ...

2

You seem to be mixing two very different concepts here. Quantum cloning is talking about the absolute limits of what is theoretically possible in a perfect world. In this absolute theoretical limit, yes we can derive how well quantum cloning can work, and we also know that classical cloning is nominally perfect. There is then a separate question of how well ...

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