In order to rotate about an axis of the Bloch sphere we ususally use pulses e.g. in trapped ion quantum computing or superconducting qubits. Let's say we have rotation around the x-axis. What do I have to change in order to be able to rotate around the y-axis or the z-axis? I assume it has something to do with the phase but I could not find a good reference for how this works.
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1$\begingroup$ what kinds of operations are you allowing? If for example you can apply an Hadamard gate then rotations around X can be converted into rotations around Z, and the other way around $\endgroup$– glS ♦Jun 4, 2018 at 15:38
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$\begingroup$ Unfortunately, I don't know how to realize a Hadamard gate in practice (e.g. with superconducting qubits) but this might be starting point. $\endgroup$– QuasarJun 5, 2018 at 9:55
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$\begingroup$ hence my asking what operations you are allowing/you have available $\endgroup$– glS ♦Jun 5, 2018 at 10:24
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2$\begingroup$ I have found the answer to my question. The trick is to add a $\pi/2$ phase shift (with respect to the x-rotation) to the pulse. This then also allows us to implement e.g. Hadamard gates like this: $H = e^{i\pi/2} R_x(\pi) R_y(\pi/2)$. The angle for each rotation is set by choosing the time of the pulse. $\endgroup$– QuasarJun 5, 2018 at 12:20
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4$\begingroup$ if you have available rotations are X and Y axes then sure, that works. Indeed, with X and Y rotations you can make any possible one-qubit unitary. Note that you can write an answer to your own question. $\endgroup$– glS ♦Jun 5, 2018 at 12:58
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2 Answers
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Whilst we normally talk about $\left|0\right>$ and $\left|1\right>$ as unchanging states in quantum computing, this is not usually the case in a physical realization where there tends to be an energy difference $\Delta E$ between these states such that $\left|1\right>_\mathrm{logical} = e^{-it \Delta E / \hbar} \left|1\right>_\mathrm{physical}$. This phase rotation has the same angular frequency $\Delta E / \hbar$ that you need to drive a transition between $\left|0\right>$ and $\left|1\right>$. Their relative phase is the angle that determines if you drive a $X$- or $Y$-rotation (or rotate about an axis somewhere else in the plane spanned by the axes for $X$- and $Y$-rotations).
The easiest way to achieve a $Z$-rotation is to wait and let the $e^{-it \Delta E / \hbar}$ factor achieve it for you. However, this would be a $Z$ gate on every qubit which is not the goal if you want a $Z$ gate on one spcific qubit. To achieve that, you can for example combine partial rotations about the axes for $X$- and $Y$-gates. Its easy to verify with a globe (or any ball with marked directions) that "half" of an $X$ gate (a 90 degree rotation) followed by a $Y$ gate and then the reverse "half" if an $X$ gate (a -90 degree rotation) together is the same as a $Z$ gate.
There are more complicated tricks you could alternatively use. For example, by driving a pulse in a detuned way, you move the effective Bloch sphere rotation axis away from the equatorial plane (in which the rotation axes for $X$ and $Y$ gates lie). But this works only for moderate tiltings of this angle as the pulses' effect in the case of trying to realize a $Z$ gate this way would tend to zero: It happens infinitely slow. Hence you need to combine at least two Bloch sphere rotations to get a Z gate with such pulses driving the transition between your qubit basis state.
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For superconducting qubits, x and y rotations are usually both done with microwave pulses, and as you said the phase of the pulse determines the rotation axis. See mathematical details in this Physics Stack Exchange post: How do we perform transverse measurements in a two level system?
Rotations about the z axis are quite different; they are done by changing the qubit's resonance frequency (a.k.a. "detune") for a specified duration of time. For example, detuning by 1 MHz for 100 ns gives a z-axis rotation by 1/10 of a full rotation.
