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.

💡 Research Summary

The paper studies a quantum analogue of the classical k‑XOR problem, namely k‑XOR Hamiltonians, which are signed sums of k‑local Pauli operators. The authors focus on two average‑case models: (i) a semirandom model where the hypergraph of interactions is arbitrary but the signs b_C are independent Rademacher variables, and (ii) a fully random model where the signs are i.i.d. standard Gaussian. Their main contribution is a classical spectral algorithm that, given such a Hamiltonian, computes in time n^{O(ℓ)} a number algval(H_I) that upper‑bounds the maximum eigenvalue λ_max(H_I). When the number of terms satisfies |H| ≥ O(n)·⌈n/ℓ⌉^{k/2−1}·log n·ε^{−4}, the algorithm certifies that λ_max(H_I) ≤ ½ + ε with high probability. This matches the well‑known refutation threshold for random k‑XOR CSPs and extends it to the quantum setting.

The algorithm is built on a quantum version of the Kikuchi matrix hierarchy. First, the expectation ⟨ψ|H_I|ψ⟩ is expressed as a quadratic form of a lifted Kikuchi matrix K̃(ℓ) whose rows and columns correspond to degree‑ℓ Pauli operators. For even k the standard Kikuchi construction suffices; for odd k the authors introduce a bipartite reduction, an odd‑arity Kikuchi matrix, and an edge‑deletion procedure to handle the asymmetry. The second step bounds the spectral norm of K̃(ℓ) via the trace‑moment method, exploiting the independence and zero‑mean of the random signs to apply matrix Chernoff bounds. The resulting bound on ‖K̃(ℓ)‖ yields the desired certification guarantee.

In parallel, the paper establishes matching lower bounds for the non‑commutative Sum‑of‑Squares (ncSoS) hierarchy. For “one‑basis” Hamiltonians—those whose Pauli terms are simultaneously diagonalizable—the authors show that degree‑ℓ ncSoS cannot certify a value below ½ + ε when the same term count condition holds. The key technical tool is a reduction (Theorem 1.5) that transforms any classical k‑XOR instance into a one‑basis Hamiltonian by replacing each Boolean variable with a Z‑Pauli operator while preserving both the optimum value and the SoS value. Consequently, any classical SoS‑hard instance (e.g., those from