# How to benchmark approximate random unitary sampling

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 $$\hat{H}(t)=\hat{H}_0+\hat{H}_d(t)$$

$$$$\begin{gathered} \hat{H}_0=\sum_{i=1}^L h_i \hat{\sigma}_i^z+B_0 \sum_{i=1}^L \hat{\sigma}_i^x+J \sum_{i=1}^{L-1} \hat{\sigma}_i^z \hat{\sigma}_{i+1}^z, \\ \hat{H}_d(t)=\delta B \cos (\omega t) \sum_{i=1}^L \hat{\sigma}_i^x . \end{gathered}$$$$

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 :

$$$$\hat{U}=\hat{\mathcal{T}} \exp \left[-i \int_0^T \hat{H}(t) d t\right] \equiv \exp \left[-i \hat{H}_F T\right]$$$$

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

$$$$\text { LXEB }:=\sum_{j=1}^k\left|\left\langle s_j\left|C\right| 0^n\right\rangle\right|^2$$$$

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.

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

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

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:

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

And the distance to $$k$$-design can be computed as $$$$\mathcal{F}_{\mathcal{E}}^{(k)}-\mathcal{F}^{(k)}_{U(N)}$$$$ As one could have notice, matrices of the COE, have an additional orthogonality constraint, so for \ref{eq::framepor} to make sens, elements of any $$\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

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

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

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