Let $J$ be the $n$ by $n$ matrix of all 1's. Let $f(n)$ be the least number $m$ of unitary matrices $U_1,\dots,U_m$ so that $J = U_1 + \cdots + U_m$. What can you say about the growth of the function $f(n)$?
On sums of unitary matrices
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linear-algebra
matrices
optimization