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I have some questions regarding the mechanism of Qiskit pulse.ShiftPhase instruction:

  1. Does it work like a Phase Shift $P(\theta)=\begin{bmatrix} 1 & 0 \\0 & e^{i\theta} \end{bmatrix}$ gate, a Global Phase Shift $Ph(\theta)=\begin{bmatrix} e^{i\theta} & 0 \\0 & e^{i\theta} \end{bmatrix}$ gate, or a $R_z(\theta)=\begin{bmatrix} e^{-i\theta/2} & 0 \\0 & e^{i\theta/2} \end{bmatrix}$ gate?
  2. I read that when I apply a ShiftPhase it'd affect all the pulses following, meaning everytime I apply a pulse there will be a phase shift gate right? So if I only want to apply only one RZ/PS/Global PS gate should I apply a reversal phase shift after a pulse, or would that equivalently mean I have applied a PS/GPS/RZ($ -\theta$) gate?
  3. How is the free Z rotation tracked by software in the case of a qutrit (3 energy levels) or more, and how does ShiftPhase work in this case?

Many thanks!

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First note that your $P(\theta)$ and $R_z(\theta)$ gates are the same up to a global phase, so you will not be able to distinguish them.

Now, according to qiskit.pulse.ShiftPhase documentation:

The qubit phase is tracked in software, enabling instantaneous, nearly error-free Z-rotations by using a ShiftPhase to update the frame tracking the qubit state.

So the ShiftPhase pulse implements a $R_z(\theta)$ gate (or $P(\theta)$ as they are the same).

If you want to shift the phase only for one gate, you will have to apply a $P(-\theta)$ (via ShiftPhase) just after the gate. It is likely that the ShiftPhase will need a positive phase, as it should increase the phase according to the documentation:

The ShiftPhase instruction causes $\phi$ to be increased by the instruction’s phase operand.

In this case, you can use $P(-\theta) = P(2\pi - \theta)$ and still apply a ShiftPhase.

About your third question, the software (qiskit here?) is not aware of qutrits. It just "attach" to every pulse a given phase, pre-computed when the quantum circuit is transformed to a Schedule, and executes each pulse with its attached phase. There is no notion of $n$-level state here, it just impact the pulses. If your pulses explore a $3$-level state then so be it, but I do not think the phase-tracking part changes in any way.

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