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I'm going through "Quantum Computation & Quantum Information" by Michael A. Nielsen and Isaac L. Chuang, and as a high school student with no previous knowledge, I cannot understand some things about linear transformations on the bloch sphere.

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  • $\begingroup$ Hi, and welcome to QCSE. What do you understand so far? Do you know what the $X$ gate does? Can you visualize the Bloch sphere in your mind's eye? Can you see how $\vert 0\rangle$ and $\vert 1\rangle$ correspond to poles of the sphere? Can you imagine sticking a line through the $Y$ axis, and giving the sphere a $180^\circ$ rotation? $\endgroup$ – Mark S May 10 at 18:17
  • $\begingroup$ I visualise X and Z gates but cannot do the same with Y, because I dont have deep understanding of complex numbers. Can you give me a simple explanation? $\endgroup$ – Jimarious May 10 at 20:11
  • $\begingroup$ Hi Dimitris, a great place to start if you are new is grab a copy of Susskinds' Quantum Mechanics the theoretical minimum. This book explains all the key concepts and required mathematical concepts in an easy to access no bs manner to get you going into QCQI $\endgroup$ – Sam Palmer May 10 at 20:55
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The nice thing about the Bloch sphere is that you can visualize the various rotations without having to think too much about complex numbers.

The $Y$ gate causes a 180-degree rotation around the $y$-axis of the sphere. (Imagine spinning the sphere a half-rotation around the $y$-axis.) For example, if your state is at either end of the $x$-axis, then rotating 180 degrees around the $y$-axis will move your state to the opposite end of the $x$-axis. Same if your state is at either end of the $z$-axis. But if your state is on the $y$-axis itself, then applying a $Y$ gate does not rotate the state — it remains in the same place.

Mathematically, yes, this is represented with complex numbers. If it helps, you can think of the $x$-$y$ plane of the Bloch sphere as sort of like polar coordinates representing the relative phase between the $\left|0\right>$ and $\left|1\right>$ parts of your superposition — the $x$-axis represents the real part, and the $y$-axis represents the imaginary part.

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