It has been known for almost three decades that many $\mathrm{NP}$-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit $C$ with $n$ uninitialized inputs, $\mathit{poly}(n)$ gates, and treewidth $t$, one can compute in time $(\frac{n}��)^{\exp(O(t))}$ a classical assignment $y\in \{0,1\}^n$ that maximizes the acceptance probability of $C$ up to a $��$ additive factor. In particular, our algorithm runs in polynomial time if $t$ is constant and $1/poly(n) < ��< 1$. For unrestricted values of $t$, this problem is known to be complete for the complexity class $\mathrm{QCMA}$, a quantum generalization of MA. In contrast, we show that the same problem is $\mathrm{NP}$-complete if $t=O(\log n)$ even when $��$ is constant. On the other hand, we show that given a $n$-input quantum circuit $C$ of treewidth $t=O(\log n)$, and a constant $��<1/2$, it is $\mathrm{QMA}$-complete to determine whether there exists a quantum state $\mid\!��\rangle \in (\mathbb{C}^d)^{\otimes n}$ such that the acceptance probability of $C\mid\!��\rangle$ is greater than $1-��$, or whether for every such state $\mid\!��\rangle$, the acceptance probability of $C\mid\!��\rangle$ is less than $��$. As a consequence, under the widely believed assumption that $\mathrm{QMA} \neq \mathrm{NP}$, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.

30 Pages. A preliminary version of this paper appeared at the 10th International Computer Science Symposium in Russia (CSR 2015). This version has been submitted to a journal and is currently under review