Spectral Certificates and Sum-of-Squares Lower Bounds for Semirandom Hamiltonians

The $k$-$\mathsf{XOR}$ problem is one of the most well-studied problems in classical complexity. We study a natural quantum analogue of $k$-$\mathsf{XOR}$, the problem of computing the ground energy of a certain subclass of structured local Hamiltonians, signed sums of $k$-local Pauli operators, which we refer to as $k$-$\mathsf{XOR}$ Hamiltonians. As an exhibition of the connection between this model and classical $k$-$\mathsf{XOR}$, we extend results on refuting $k$-$\mathsf{XOR}$ instances to the Hamiltonian setting by crafting a quantum variant of the Kikuchi matrix for CSP refutation, instead capturing ground energy optimization. As our main result, we show an $n^{O(\ell)}$-time classical spectral algorithm certifying ground energy at most $\frac{1}{2} + \varepsilon$ in (1) semirandom Hamiltonian $k$-$\mathsf{XOR}$ instances or (2) sums of Gaussian-signed $k$-local Paulis both with $O(n) \cdot \left(\frac{n}{\ell}\right)^{k/2-1} \log n /\varepsilon^4$ local terms, a tradeoff known as the refutation threshold. Additionally, we give evidence this tradeoff is tight in the semirandom regime via non-commutative Sum-of-Squares lower bounds embedding classical $k$-$\mathsf{XOR}$ instances as entirely classical Hamiltonians.