Quantum Kernels and the Cross-Section of Stock Returns: Anatomy of a Vanishing Advantage

Do quantum kernels improve cross-sectional stock return prediction? We run a controlled horse race on the Chinese A-share market in which a quantum fidelity kernel, a projected quantum kernel, and a classical RBF control share identical training subsamples, solver, and tuning budgets, so that only the kernel is exchanged. On the main evaluation -- a point-in-time universe and 170 walk-forward windows (2012-2025) -- no quantum advantage exists: the fidelity kernel is indistinguishable from its RBF control ($\Delta$IC $=+0.005$, $p=0.42$), and a $2\times2$ design crossing kernel type with training budget (a Nystrom extension to the full ~38,000-observation windows) shows quantum kernels matching, but never beating, equal-budget linear models; after family-wise correction no pairwise difference among eleven models is significant, with point estimates favoring penalized linear regressions throughout. We then document how the opposite conclusion arises: a 60-window evaluation on a universe screened with full-sample information makes the same quantum kernel appear dominant on stability criteria and significantly better than neural baselines. Interaction characteristics from the anomalies literature help nothing, quantum or classical; a widened bandwidth grid reveals an interior optimum rather than the near-classical endpoint a coarse grid suggests; and the geometric difference, while large throughout ($g \gg 1$), does not predict out-of-sample gains ($\rho=-0.20$). We propose protocol standards -- kernel-swap controls, budget-equalized comparisons, point-in-time universes, and multiplicity-robust inference -- for empirical claims of quantum advantage in finance.