Tensor network non-zero testing

Tensor networks are an important tool in condensed matter physics. In this paper, we study the task of tensor network non-zero testing (\tnit): Given a tensor network $T$, does $T$ represent a non-zero vector? We show that \tnit~is not in the Polynomial-Time Hierarchy unless the hierarchy collapses. We next show (among other results) that the special cases of \tnit~on non-negative and injective tensor networks are in NP. Using this, we make a simple observation: The commuting variant of the MA-complete stoquastic $k$-SAT problem on $D$-dimensional qudits is in NP for $k\in O(\log n)$ and $D\in O(1)$. This reveals the first class of quantum Hamiltonians whose commuting variant is known to be in NP for all (1) logarithmic $k$, (2) constant $D$, and (3) for arbitrary interaction graphs.