Which Schur sampling and transform tasks can STP implement?

Explicit variants

1. Information in a nonadaptive record

   For an unknown $n$-qubit state, fix a distribution over ordered pair-measurement schedules independent of outcomes. Characterize which functions of its Schur–Weyl blocks can be estimated from the complete schedule and outcome record. Specify whether each trial uses a fresh copy, and bound copy and measurement complexity at additive error $\varepsilon$.

2. Optimal sampling and invariant estimation

   For weak Schur sampling, unitary Schur sampling, and a specified invariant observable, determine optimal STP measurement and copy complexity as functions of $n$ and $\varepsilon$. Define the required classical and retained quantum outputs separately. Compare adaptive protocols with nonadaptive random-SWAP protocols under the same input and error conventions.

3. A full Schur transform

   Can STP implement an encoded coherent Schur transform, or a specified measurement-based analogue, at lower physical cost than a conventional gate decomposition? Specify how the $J$, magnetic, and multiplicity registers are represented and read out. A coherent version must preserve their correlations with an external reference; compare measurement count, adaptive depth, ancillas, and error at fixed input/output conventions.

4. A Schur–Weyl correlation problem

   Define two efficiently specified representation-theoretic descriptions and a promise about their correlation, with matched classical and quantum oracle access. Find a polynomial-query STP algorithm and a superpolynomial classical query lower bound. The basis changes, state preparation, and any coherent extraction used by the algorithm count as resources.