We know that $T$ gate is as follows

$T=\left(\begin{array}{cc}1 & 0 \\ 0 & e^{i \pi / 4}\end{array}\right)$

And on the other resultant spin-resonance hamiltonian is

$\frac{H}{\hbar}=\frac{1}{2} \omega_{0} Z+g(X \cos \omega t+Y \sin \omega t)$

And if we go into the rotating frame we will have

$\mathrm{i} \frac{\partial}{\partial t}|\varphi(t)\rangle=\underbrace{\left[\mathrm{e}^{\mathrm{i} \omega t Z / 2} \frac{H}{\hbar} \mathrm{e}^{-\mathrm{i} \omega t Z / 2}-\frac{\omega_{0}}{2} Z\right]}_{\left[\frac{\omega_{0}-\omega}{2} Z+g X\right]|\varphi(t)\rangle}|\varphi(t)\rangle$.

So for example we can construct NOT operation, if the frequency of the AC magnetic field ($\omega$) is equal to the larmor frequncy ($\omega_0$), and then we have only $X$ gate.

Notice that $\omega$ is adjustable and using that we want to achieve $T$ gate. What do you think about that?


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