Which computational tasks does Schur–Weyl duality relate?

Explicit variants

1. Sampling simulation

   Compare sampling a final coupling-tree label in the weak permutational model with sampling an STP outcome record. Specify allowed initial labels, a polynomial-time input description, the output map, and total variation error.

2. Amplitude estimation

   For specified input/output coupling-tree basis states and a given permutation, compare additive-precision transition-amplitude estimation with an STP task. Sampling probabilities alone does not provide phase-sensitive amplitudes.

3. Coherent simulation

   Require a transformation on encoded superpositions that preserves correlations with an external reference, with a stated diamond-norm error. Identify the extra recoupling or coherent-control operations required.

4. Decision power

   After fixing each model’s uniformity, preparation, readout, and bounded-error convention, compare the resulting language classes. Equality of commuting algebras does not establish equality of these operational models.

5. Fourier-type extraction and coherent recoupling

   Specify black-box or circuit access to collective rotations or permutations. Which coherent irrep-extraction tasks follow, at what precision and cost? Separately ask for an STP implementation of a coupling-tree basis change on a multiplicity register, preserving an external reference. Existing weak permutational sampling simulation does not settle this coherent task.

6. What representation data can be measured?

   Distinguish Young shape, nested tableau labels, characters of specified permutations, and recoupling amplitudes. State the input representation, permitted operations, number of copies, required precision, and tolerated disturbance for each task.

7. Transferring information between the two factors

   Within $V_J\otimes M_J$, can a specified encoding transfer a logical state between an $\mathrm{SU}(2)$ irrep register and an $S_n$ multiplicity register, preserving an external reference? The registers need not have equal dimensions, so fix equal-dimensional code spaces and account for ancillas. On a closed fixed-spin register an STP branch acts as $I_{V_J}\otimes K$, which obstructs unrestricted transfer from $V_J$ by STP alone. Determine what extra operations or relational encodings permit a useful Fourier-like correspondence.

8. Algorithms in dual representation coordinates

   Choose a benchmark such as order finding, a quantum walk, or a correlation problem. Give two precise representation-theoretic descriptions and the transformation between them; identify the operations simplified by each description. Evaluate the resulting algorithm or explanation against a specified gate or query baseline, including the cost of moving between descriptions.