We introduce Hypergraph Cycle Enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference tasks. The study focuses on a representative case, planted noisy $k$-XOR (also known as sparse Learning Parity with Noise, LPN). Both theoretical analysis and experiments show that HCE attains recovery performance comparable to the Kikuchi method, matching the Kikuchi hierarchy. We identify regimes where HCE outperforms Kikuchi, demonstrating that neither algorithm dominates universally.\ Moreover, by replacing one step of the algorithm with a quantum walk, we obtain a sub‑quadratic quantum speedup. This result does not refute the quartic quantum speedup of the quantum Kikuchi method, which remains valid in the practically relevant regime of large size and high noise strength.\ \ Review