Towards Optimal Quantum Estimators for State Frame Potential
The state frame potential is a standard diagnostic of how closely a quantum state ensemble approximates Haar randomness. In this work, we study the problem of estimating the state frame potential of order $t$ to within additive error $\varepsilon$ under three progressively weaker access models: (i) query access to a multi-state-preparation oracle, (ii) general sample access, and (iii) single-copy sample access. In the query model, we establish a near-optimal query complexity of $\widetilde{\Theta}(\sqrt{t}/\varepsilon)$, yielding a quadratic improvement in the dependence on $t$ over the previous best result of Nakata, Takeuchi, Kliesch, and Darmawan (PRX Quantum 2025). In the general sample model, we establish the optimal sample complexity $\Theta(t/\varepsilon^2)$. In the single-copy sample model, we present a store-and-estimate approach whose sample complexity depends on the R\'enyi entropy of the ensemble weights. As an application, we use the single-copy algorithm to assess the randomness of projected state ensembles, where the entropy term becomes the observational R\'enyi entropy associated with measuring one subsystem.