9 problems

Kong–Li–Liu's iSWAP optimality conjecture. For a general fixed connected graph, Kong, Li, and Liu conjectured that $\mathrm{iSWAP}$ maximises the Hermitian second-moment spectral g…

Kong–Li–Liu's ironed-gadget degeneracy conjecture. Every ironed gadget with KAK parameter $a=5/9$ has the same second-moment spectral gap as $\mathrm{iSWAP}$.

Consider local Haar-random gates acting on a many-body system and a global Haar-random gate on the corresponding total Hilbert space. Asymptotic equivalence conjecture. In the limi…

Linear-graph connection optimality conjecture. No other graph on $n$ qudits requires more connections to form an $\epsilon$-approximate $t$-design than the linear graph, which requ…

Let $G_{\star}$ be a star graph with $n_{\star}$ vertices, local dimension $q$, and moment $k$, and let $\Delta(G_{\star},n_{\star},k)$ denote its spectral gap. Higher-moment star-…

For a star graph $G_{\star}$ with $n_{\star}$ vertices, let $\Delta(G_{\star},n_{\star},k)$ denote the spectral gap of its Hamiltonian at moment $k$. Low-moment star graph conjectu…

Consider the one-dimensional CQA ensemble on $n$ qudits with SU$(d)$ symmetry, and let $\epsilon>0$ denote the target approximation precision. CQA convergence conjecture. Based on…

Let $n$ qubits be equipped with a 2-local one-dimensional CQA random circuit, and let $\theta$ denote the circuit depth. For $\epsilon>0$, an ensemble is an $\epsilon$-approximate…

Linear-depth conjecture. For the general random quantum circuit model, a depth of order $O(nt)$ suffices to approximate $t$-designs.