# What are efficient methods for measuring gate noise?

In this question I'm concerned with getting detailed spectral information about analog noise in quantum gates.

Assume we have imperfect control over transitions between $$\vert 0\rangle$$ and $$\vert 1\rangle$$ states of a qubit (i.e. an imperfect single qubit gate). In particular, we may control the amplitude, phase, and duration of the Rabi drive $$\Omega(t)=\Omega_0(t) \cos(\omega t+\phi)$$ between the qubit states. In matrix form the net Hamiltonian interaction generally looks like:

$$H_{int}=\begin{pmatrix} 0 & \Omega(t)\\ \Omega^{*}(t) & 0 \\ \end{pmatrix}$$

For now, I assume the qubit is perfect, but the drive $$\Omega(t)$$ is imperfect.

By imperfect drive, I mean that the amplitude has a noise contribution $$\Omega_0+\delta\Omega(t)$$, and similarly for the phase $$\phi=\phi_0+\delta\phi(t)$$. To practically understand this noise, it is more meaningful to treat it in the frequency-domain $$\delta\Omega(f)$$ and $$\delta\phi(f)$$. Such noise may be caused spurious microwave pickup or by vibrations at particular frequencies, for example.

The question I am interested in is what are experimental methods to measure the amplitude $$\delta\Omega(f)$$ and phase $$\delta\phi(f)$$ noise spectra of your gate through quantum mechanical measurements of your qubit?

Some notes

1. I am imagining using the qubit to measure the noise, not some classical external gadget (like an oscilloscope etc).

2. There are techniques for measuring the total gate error using some sort of randomized testing, but no insightful information about the noise is obtained that way.

3. I expect spin echo sequences should be useful here, but they don't seem efficient at differentiating requencies. Is there a scheme to measure one of these noise quantities while being sure they are insensitive to noise in the other? It also seems unclear how to verify the measurement is linearly sensitive to noise.