Scaling and constants of weak Schur sampling under limited memory

Determine the precise sample-complexity scaling laws and constant factors for weak Schur sampling when used for hypothesis testing of unitarily invariant properties of quantum states, with particular emphasis on low-rank states and implementations constrained by limited working memory (e.g., logarithmic-size quantum memory).

Background

The paper studies incoherent sensing problems where only unitarily invariant properties of an unknown state are assumed known. It shows that weak Schur sampling (WSS) is minimax-optimal for distinguishing such composite hypotheses and provides sample-complexity results for rank, purity, and spectral-gap testing that avoid dimension dependence.

While asymptotic scalings are given for several tasks (e.g., rank testing Θ(r_n^2 log(1/β)/θ_0)), the authors note that a complete characterization of scaling behavior and constant factors for WSS—especially in the practically relevant regimes of low-rank states and limited working memory—remains unresolved. Tight constants and refined scaling laws would clarify the true advantages of WSS over tomography and other methods in resource-constrained settings.