Linear-graph connection optimality conjecture for approximate unitary designs

Let $n$ qudits be arranged according to a graph, and let an $\epsilon$-approximate $t$-design be formed by a random quantum circuit using two-qudit gates on the graph's edges. The connection count is the number of graph connections required to form the design.

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 requires

$$\Theta(\log n)$$

connections.

Numerical results indicate that the tested graph families require roughly comparable connection counts, with the linear graph appearing slowest among the tested architectures. The authors do not test every possible graph or values $t>2$, so the conjecture remains speculative.