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Just some more information on $T_1$ and $T_2$ times: $T_1$ and $T_2$ are known as two measurements of decoherence. In general, $T_1 > T_2$. $T_1$ = Relaxation Time = Average Transition Time between two states. For example, the time it takes for a $|1\rangle$ state to become a $|0\rangle$ state. Imagine the Bloch sphere, the arrow pointing at the north ...

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The orthonormal basis $|j\rangle$ of the $d$ dimensional finite Hilbert space corresponds to a configuration space of equally spaced clockwise ordered $d$ points on a circle $S^1$ or equivalently, the vertices of a $d$-dimensional regular polygon. One may think of a point as a discrete location of a particle, then the shift operator $X$ shifts the particle ...

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The shift operator takes his name from the fact that it shifts the position of its input, as in, it sends $1\to2$, $2\to3$ etc, with the last computational basis element being sent back to the first one: $d\to 1$ (or the same thing starting with $0$, depending on notation). As per the "boost" operator $Z$, I have usually seen those referred to as "clock ...

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