# How can I achieve single-qubit $T$ operation using spin resonance?

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?