I'm currently studying a specific sampling "quantum advantage" (sorry for the buzzword) protocol wich consist of periodically driving a random Ising chain (https://iopscience.iop.org/article/10.1088/2058-9565/acbd69), the resulting evolution operator at stroboscopic time should approximate matrices of the Circular Orthogonal Ensemble (COE).
Preliminary on the periodically driven quantum Ising model framework:
Consider the hamiltonian of a random periodically driven Ising chain $\hat{H}(t)=\hat{H}_0+\hat{H}_d(t)$
\begin{equation} \begin{gathered} \hat{H}_0=\sum_{i=1}^L h_i \hat{\sigma}_i^z+B_0 \sum_{i=1}^L \hat{\sigma}_i^x+J \sum_{i=1}^{L-1} \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z, \\ \hat{H}_d(t)=\delta B \cos (\omega t) \sum_{i=1}^L \hat{\sigma}_i^x . \end{gathered} \end{equation}
The randomness coming form $h_i$ being uniformly distributed in the interval $[-W/2,W/2]$, $W>0$ being some disorder parameter.
Thanks to Floquet Theorem the evolution operator $\hat{U}$ can be characterise as :
\begin{equation} \hat{U}=\hat{\mathcal{T}} \exp \left[-i \int_0^T \hat{H}(t) d t\right] \equiv \exp \left[-i \hat{H}_F T\right] \end{equation}
And under the right regime of parameters $\{\hat{U}\}\approx\{U_{COE}\}$ the ensemble of circular orthogonal matrices.
Preliminary on quantum sampling advantage benchmarking :
Sampling quantum advantage experiments needs benchmarking, in the case of RCS, this is done through Linear Cross Entropy Benchmarking
\begin{equation} \text { LXEB }:=\sum_{j=1}^k\left|\left\langle s_j\left|C\right| 0^n\right\rangle\right|^2 \end{equation}
For $k$ sample of the experimental circuit, approximating the ideal circuit $C$. The point to higligh for next part is :
- When realizing such an experiment, we know wich circuit is suppose to be implemented in the physical machine,and basically, $\text{LXEB}$ evaluate the quality of this implementation subject to noise ...
Main hurdle
My main problem is the fact that in the context of COE-sampling with floquet operator $\hat{U}$, the approximation already occur before any physical implementation. In the sense that the ideal evolution $\hat{U}$ is already some kind of approximation of a typical matrix $U_{COE}$. Benchmarking similar to $LXEB$ seem impossible in this context since we don't know wich COE matrix is approximated by $\hat{U}$ at each go.
My main interrogations being :
What kind of benchmarking would be possible to characterise the approximation of the ensemble of $\{\hat{U}\}$ to the COE ensemble (some kind of COE approximate t-design benchmarking?).
How such benchmarking can be relevant in characterizing the actual hardness of sampling ${\hat{U}}$, (the hardness of sampling typical COE matrices being adressed in the paper linked above).
Hope this is not to fuzzy.