Limitations and Separations in the Quantum Sum-of-squares, and the Quantum Knapsack Problem

We answer two questions regarding the sum-of-squares for the SYK model left open in Ref. 1, both of which are related to graphs. First (a "limitation"), we show that a fragment of the sum-of-squares, in which one considers commutation relations of degree-$4$ Majorana operators but does not impose any other relations on them, does not give the correct order of magnitude bound on the ground state energy. Second (a "separation"), we show that the graph invariant $\Psi(G)$ defined in Ref. 1 may be strictly larger than the independence number $\alpha(G)$. The invariant $\Psi(G)$ is a bound on the norm of a Hamiltonian whose terms obey commutation relations determined by the graph $G$, and it was shown that $\alpha(G)\leq \Psi(G) \leq \vartheta(G)$, where $\vartheta(\cdot)$ is the Lovasz theta function. We briefly discuss the case of $q\neq 4$ in the SYK model. Separately, we define a problem that we call the quantum knapsack problem.