Suppose we have a finite set of positive reals $c_i$ such that $\sum_i c_i = 1$. Define the corresponding unit sum of a set of matrices $M_i$ to be $\sum_i c_i\cdot M_i$.
Does there exist a set of $c_i$ and a set of (more than one!) distinct unitary $M_i$ such that the unit sum is also unitary? If so, what's the smallest matrix dimension for which such a set exists?
Lemma. Separating any unitary matrix as $U= A+iB$ where $A$ and $B$ are real, one sees that each column $A_j$ has length at most one.
Proof. Since $I=U^*U = (A^t-i B^t)( A+i B) = A^t A + B^t B + i(A^tB- B^tA) \iff A^tA+ B^tB=I$ and $ A^tB=B^tA$. Thus from the $(j,j)$ entry of the identity $A^tA+B^tB=I$ we see that the $j_{th}$ columns of $A$ and $B$ are related by $||A_j||^2+ ||B_j||^2=1$.
Suppose now that a nontrivial convex combination of unitary matrices is unitary: $U=\sum_k c_k U_k$. After multiplying across by $U^{-1}$ deduce that $$I= \sum_k c_k \tilde U_k$$ for some new set of unitary matrices $\tilde U_k$.
Extract the real part of this relation to obtain $I= \sum_k c_k\tilde A_k$.
Each column of this latter identity asserts that each column of $I$, which is a vector on the unit sphere, is expressed as a convex combination of vectors that each have length at most one. Since the sphere is strictly convex, there can be no such nontrivial convex combination.
This is a simple modification of a previous argument in a related post: Can a rotation matrix be written as the convex combination of two rotation matrices?