Quantum ordinal selection with two comparisons

Abstract

We study the optimal probability of selecting an object within a prescribed rank threshold using at most two noiseless coherent pairwise comparisons of an unknown strict order. Polynomial degree bounds alone can overestimate the success attainable with two comparisons. We develop a comparison-specific semidefinite model that completely characterizes attainable success. Computer-assisted proofs establish the exact optimum for every rank threshold at seven objects and for minimum selection at eight. For minimum selection, we construct an explicit algorithm family and determine its exact optimum. Bounds for all admissible two-comparison algorithms show that the family's asymptotic success is within 0.016% of the unrestricted optimum. We also prove that this relative gap has a strictly positive limit. The optimal quantum-to-classical success ratio for minimum selection converges to approximately 2.48 under the same two-comparison budget.