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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.

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One approach to measure gate noise (error) is interleaved randomized benchmarking. You will find several references if you do a paper search but here is one reference to get started: https://arxiv.org/pdf/1504.06597.pdf

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    $\begingroup$ I'm aware, but that cannot be used to obtain noise spectral information, it only really tells you about the net noise (aka gate error) $\endgroup$ Commented Oct 31, 2023 at 21:28
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    $\begingroup$ To emphasize, I am making a distinction between the (net) gate error and the gate noise spectrum here. $\endgroup$ Commented Oct 31, 2023 at 21:29
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As far as I know, qubit operations are well-trusted (since gate fidelities on leading platforms are >99%) and drives (especially microwave drives) can be well-characterized by external classical devices like an oscilloscope or spectrum analyzer without the complications that arise from projection noise of a quantum measurements. The field of quantum optimal control is built on the fact that qubit control can be controlled to such a high degree that gate operations can be designed to compensate for qubit imperfections.

It's not exactly what you're asking for, but there is a related field of (quantum) noise spectroscopy that explores how qubit interactions with a bath can be described with a noise spectrum, and how the amplitude and phase of that qubit's coherence under different spin echo sequences encode different properties of the bath.

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