The iSWAP optimality conjecture for 2-local unitary circuit ensembles

Let $n\geq 3$, let $\mathcal{G}$ be a fixed connected graph, and let $\mathcal{E}$ be a 2-local unitary circuit ensemble. Write $\operatorname{\mathcal{T}}_{2,\mathcal{G}}^{\mathcal{E}}$ for its associated second-moment operator and $\Delta(\cdot)$ for the convergence gap. Assume that $\operatorname{\mathcal{T}}_{2,\mathcal{G}}^{\mathcal{E}}$ is Hermitian. The iSWAP optimality conjecture.

$$\Delta\!\left(\operatorname{\mathcal{T}}_{2,\mathcal{G}}^{\mathcal{E}}\right)\leq \Delta\!\left(\operatorname{\mathcal{T}}_{2,\mathcal{G}}^{\operatorname{\mathsf{iSWAP}}}\right).$$

The conjecture asserts that the iSWAP gadget gives at least as large a convergence gap as every admissible 2-local circuit ensemble on every connected interaction graph, supporting its optimality in forming unitary 2-designs. The paper states that this claim has not been fully resolved.