Planted Cliques and Quantum Symmetry-Adapted Measurements
We study how quantum encodings and symmetry-adapted measurements preserve information for planted-clique detection from one classical graph. For $k=\lfloor n^{1/2-\varepsilon}\rfloor$, with fixed $0<\varepsilon<1/2$, detection is statistically possible but conjectured hard for polynomial-time classical algorithms. For a compact binary phase encoding, we prove that constant-advantage detection requires $Ω(n^{1+2\varepsilon}\ln^2 n)$ copies of the phase state of the same graph, even under arbitrary joint measurements. In the large-copy limit, the optimal decision rule thresholds the total number of $k$-cliques in the graph and its complement, but this characterization provides no efficient detector. We therefore explore measurements guided by the symmetries of the input distributions, starting with the efficient Schur transform on the full graph register. We show that weak Schur sampling, which measures only the representation label, depends only on edge count and has vanishing distinguishing power in this regime. When the label and multiplicity registers are discarded, the remaining quantum states are almost perfectly distinguishable. We show that the support of the planted state occupies only a vanishing fraction of the graph Hilbert space. Any subspace containing it still permits near-perfect detection if its relative dimension also vanishes. This gives us freedom to choose a subspace that is easier to measure. We propose exploring subgroup isotypic measurements to find such subspaces. Whether they can yield an efficient detector remains open. Finally, we show that a single supplied coherent quantum sample permits efficient detection, yielding a conditional computational separation from one classical sample under quantum planted-clique hardness.