# Rotating about the y- or z-axis of the Bloch sphere

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.

• 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
– glS
Jun 4, 2018 at 15:38
• Unfortunately, I don't know how to realize a Hadamard gate in practice (e.g. with superconducting qubits) but this might be starting point. Jun 5, 2018 at 9:55
• hence my asking what operations you are allowing/you have available
– glS
Jun 5, 2018 at 10:24
• 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. Jun 5, 2018 at 12:20
• 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.
– glS
Jun 5, 2018 at 12:58

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.

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.