Task-Sufficient Quantum Measurements for Certified Clustering

Quantum-kernel-based clustering methods typically estimate pairwise quantum-state fidelities to construct a similarity graph and then derive a clustering partition from the resulting graph. However, estimating pairwise fidelities separately across all sample pairs requires measurements on $\Theta (n^2)$ pairs, even though the final partition may depend on far less information. Moreover, existing methods return a partition without certifying whether the acquired measurements are sufficient to uniquely determine it. To address these limitations, we propose Quantum Certification for Clustering (Q-Cert), a task-oriented online quantum measurement framework. Q-Cert establishes within-cluster connectivity through selected overlap tests and reuses single-state measurements to exclude multiple cross-cluster edges; the resulting evidence is integrated into confirmed and possible graphs that determine when the partition can be safely certified. Theoretically, Q-Cert guarantees that the probability of incorrectly certifying a partition is at most \(\delta\) for arbitrary pure-state inputs. Moreover, as long as no pairwise fidelity lies exactly at the threshold, Q-Cert can complete certification even when the pilot and reusable evidence provide no useful information. Under suitable connectivity-witness and reusable-separation conditions, Q-Cert requires only $O(n\log (n/\delta ))$ quantum-oracle calls. On constructed pure-state families, we establish two complementary separations: partition certification avoids a gap-dependent full-graph recovery cost under identical measurement access, while informative reusable measurements yield a quadratic-to-near-linear separation for the same clustering task relative to restricted access. On the bounded-degree family, savings over full-risk direct increase over the tested sizes, reaching 75.32% at $n=192$.