Planted Cliques and Quantum Symmetry-Adapted Measurements

Abstract: The planted clique problem is a promising candidate for quantum advantage with a wide computational-statistical gap and substantial evidence for classical hardness. We study two quantum encodings of classical samples, a natural binary phase state encoding and symmetry-adapted measurements, and determine if they preserve enough information for planted-clique detection, as well as discuss their potential towards algorithmic efficiency. For the binary phase state encoding, we show that constant-advantage detection requires $Ω(n^{{1+2\varepsilon}\ln²} n)$ copies, even under arbitrary joint measurements. Measurements on $\tilde{O}(n<sup>2)$ copies suffice statistically above the logarithmic clique threshold. The symmetry-adapted measurements on the full graph register arise naturally from the Schur transform. We show that the outcome distribution of weak Schur sampling depends on the sampled graph only through its edge count and fails to distinguish the distributions; whereas retaining the representation label and Specht register after discarding multiplicity preserves distance $1-o(1)$. Near-perfect distinguishability survives even if the label is also discarded. We calculate the retained states, providing concrete targets for efficient measurement. Finally, we show that one supplied coherent quantum sample enables an efficient quantum distinguisher, which yields a conditional computational separation from one classical sample under quantum planted-clique hardness. Our results are structural and information-theoretic; efficient detection from one classical graph in the conjectured hard regime remains open.