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I am trying to use Schrodinger's equation to describe the time evolution of a qubit under a Z gate operation. For that, I need to know what the RF pulse looks like. For concreteness, let's suppose that we have a two-state atom, such as a trapped ion.

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  • $\begingroup$ could you please see if the answer below is the one your were looking for, and accept it is? $\endgroup$
    – Lior
    Feb 15, 2022 at 11:57

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Generally speaking, there are several approaches to implementing Z gates on various types of quantum computing architectures.

To understand what they are, we need to understand how XY rotations work. The basic thing to know is that in order to apply an XY gate at a specific azimuthal angle $\phi$, we start with the Hamiltonian:

$$ H= \frac{1}{2}\omega_q \sigma_z + \frac{1}{2}\Omega_R\sigma_x \cos(\omega_d t+\phi). $$

A transformation to the rotating frame gives us the Hamltonian:

$$ H=\frac{1}{2}\Omega_R(\sigma_x\cos\phi + \sigma_y \sin \phi) $$

which implements an arbitrary XY rotation, at a rate $\Omega_R$, at an angle $\phi$ from the $\hat{x}$ axis. From this, we can see that the phase of the drive field term ($\frac{1}{2}\Omega_R\sigma_x \cos(\omega_d t+\phi)$) corresponds to the Z rotation angle of the vector which lies on the XY plane. This phase is set by the first rotation, since before that rotation the qubit is in the $|0\rangle$ state and thus has no defined phase.

With this in mind, the types of Z rotations are as follows, roughly speaking:

  1. Virtual Z rotations - when the relative phase between qubits is not important, we can use the fact that frame changes and operations are equivalent in quantum mechanics, and instead of applying a pulse, simply change the phase of the next XY pulse in a cumulative way. Since we measure eventually in the Z basis, this is equivalent to applying Z rotations, but with the advantage that the fidelity of this gate is 1. However, it only works in scenarions where the two qubit interaction is of the $Z_iZ_j$ type.

  2. Composition of XY gates - you can convince yourself that it is possible to implement any Z rotation using a combination of XY gates

  3. AC stark shift - this is common in optical trapped ion schemes, such as here. We use a detuned laser drive to create an AC stark shift which rotates the qubit around the Z axis.

  4. fast bias change - this is common in flux tunable SC qubits or spin qubits. We apply a flux or charge pulse for a predetermined amount of time, which temporarily changes the level spacing and thus the phase accumulated. This is for example what is done in the google architecture.

BTW, for trapped ions, the situation is relatively complex, since there are several types of ions (Zeeman, hyperfine, optical), and the Zeeman/hyperfine qubits can be controlled using either stimulated Raman transitions or direct RF drives. In the Zeeman/hyperfine case, it is possible to use virtual Z rotations or composition of XY gates, and as we said, in the optical qubit case, typically AC stark shifts are used. (This paper can get you started if you want to explore this further). That being said, I'm not an expert on trapped ion architectures, and I am quite sure there are subtleties I might not be completely aware of regarding the Z rotations on them.

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