Complexity of the positive semidefinite matrix completion problem with a rank constraint

We consider the decision problem asking whether a partial rational symmetric matrix with an all-ones diagonal can be completed to a full positive semidefinite matrix of rank at most $k$. We show that this problem is $\text{NP}$-hard for any fixed integer $k\ge 2$. Equivalently, for $k\ge 2$, it is $\text{NP}$-hard to test membership in the rank constrained elliptope ${\text{EE}}_k(G)$, i.e., the set of all partial matrices with off-diagonal entries specified at the edges of $G$, that can be completed to a positive semidefinite matrix of rank at most $k$. Additionally, we show that deciding membership in the convex hull of ${\text{EE}}_k(G)$ is also $\text{NP}$-hard for any fixed integer $k\ge 2$.