In many textbooks (e.g. LaPierre 2021 or Byrd 2022), when dealing with the physical implementation of quantum gates in spin qubits, Larmor precession is usually introduced first as a way to rotate the state around an axis (typically $\vec{z}$) by using a static magnetic field. However, large static fields are difficult to turn on and off arbitrarily, and Larmor frequencies $\omega$ are typically very high, resulting in severe requirements on the electronics to have accurate control of the rotation. For this reason, electron spin resonance (ESR) is introduced where an oscillating field ($B_1 = cos(\omega t + \delta)$) is applied on top of the static field to induce rotation around another axis (for example, $\vec{x}$) with frequency $\Omega$. Applying the oscillating field for a time $t$ then yields a rotation of an angle $\theta = \Omega t$ around $\vec{x}$. Moreover, it is stated that by varying the phase $\delta$ of the oscillating field a rotation can be induced also on another axis (with the particular case of $\vec{y}$ rotation for $\delta = \pi/2$).

First question: does the phase $\delta$ (and thus the oscillating field $B_1$) have to be synchronized in some way to the Larmor precession? In other words, is the only thing discriminating between a $\vec{x}$ rotation and $\vec{y}$ rotation the functional form of the oscillating field ($cos(\omega t)$ or $sin(\omega t$)), irrespective of the current phase of the qubit (which is constantly changing due to Larmor precession)?

Moreover, in quantum processors such as IBM's, the control electronics is actually designed to perform just the $\vec{x}$ rotation physically. $\vec{y}$ rotations are realized by combining $\vec{z}$ rotations and $\vec{x}$ rotations, where Virtual-Z gates (McKay, 2017) are used to rotate the reference frame. In my understanding, this is equivalent to controlling the phase of the oscillating magnetic field $\delta$ (which would strenghten the idea that there is no need for any "synchronization" between the drive field and the qubit phase - the rotation axis just depends on the functional form of the drive field, irrespective of the current qubit phase). However, a couple of answers here (1, 2, with reference to the Qiskit Pulse docs) state that implementing VZ-gates in IBM's quantum hardware is made possible because the qubit phase is tracked in software. This would lead me to believe that there is indeed some synchronicity between the application of gates and the current qubit state, as if the pulses were applied by "waiting" for the state to precess e.g. by a $\pi/2$ angle naturally. However, this would strongly disagree with everything up to here.

Second question: if the rotation is irrespective of the qubit phase but depending only on the field phase, why is there any need to track the qubit phase to perform VZ gates? Moreover, tracking this phase should be extremely difficult since it is precessing at the (very high) Larmor frequencies, so how can we track it with the required accuracy to perform arbitrary rotations?

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    $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Oct 25 at 14:26


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