How to benchmark approximate random unitary sampling

Question

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.

Answer 1

I've just realized that i did not update this post despite having found som kind of answer. It is related to the notion of $k-$designs and most of the information can be foun in the following thesis: Chaos and Randomness in Strongly-Interacting Quantum Systems.

(My question is now re-oriented to the link between a measure on a space of unitaries and hardness arguments and can be found here : updated question)

I've made a short resume of the relevant part:

A unitary $k$-design is awell-knownn notion to characterise the randomness of finite sets of matrices in relation to the Haar-measure on the unitary group.

Formally a set $\mathcal{E}$ is a $k$-design if

\begin{equation} \frac{1}{|\mathcal{E}|} \sum_{U\in\mathcal{E}} P_{k,k}\left(U\right)=\int_{U(d)} P_{k, k}(U) d \mu(U) \end{equation}

Holds for all possible $P_{k,k}$, i.e for all polynomiald of degree at most $k$ in the elements of $U$ and at most $k$ on the complex conjugates of these elements. With $d\mu(U)$ the Haar-measure on $U(d)$ the unitary group.

A quantity wich measures how close a finite ensemble of unitaries is to being Haar-random is the so-called frame potential, formally defined as follow:

\begin{equation} \label{eq::framepor} \mathcal{F}_{\mathcal{E}}^{(k)}=\int_{U, V \in \mathcal{E}} d U d V\left|\operatorname{Tr}\left(U^{\dagger} V\right)\right|^{2 k}=\frac{1}{|\mathcal{E}|^2}\sum_{U,V\in\mathcal{E}}\left|\operatorname{Tr}\left(U^{\dagger} V\right)\right|^{2 k} \end{equation}

And the distance to $k$-design can be computed as \begin{equation} \mathcal{F}_{\mathcal{E}}^{(k)}-\mathcal{F}^{(k)}_{U(N)} \end{equation} As one could have notice, matrices of the COE, have an additional orthogonality constraint, so for \ref{eq::framepor} to make sens, elements of any $\mathcal{E}$ need to come from the quotient space $U(N)/O(N)$, which refers to $N\times N$ symmetric unitary matrices.

Hence by considering equivalently the Haar-measure on $U(N)/O(N)$ and it's corresponding frame potential $\mathcal{F}^{(k)}_{U(N)/O(N}$, one can compute the difference

\begin{equation} \mathcal{F}_{\mathcal{E}}^{(k)}-\mathcal{F}^{(k)}_{U(N)/O(N}\geq0 \end{equation}

Where $\mathcal{F}^{(k)}_{U(N)/O(N}=\frac{2N}{N+1}$. Moreover, the frame potential difference can be easily linked to $\epsilon$-approximate $k$-design in the 2-norm (approximate k-design are usually defined with the diamond norm). $\mathcal{E}$ is an $\epsilon$-approximate $k$-design in the 2-norm if

\begin{equation} \sqrt{\mathcal{F}_{\mathcal{E}}^{(k)}-\mathcal{F}^{(k)}_{U(N)/O(N}} \leq \epsilon \end{equation}