A memory and gate efficient algorithm for unitary mixed Schur sampling

Understand

We formalize the task of unitary Schur sampling -- an extension of weak Schur sampling -- which is the process of measuring the Young label and the unitary group register of an input $m$ qudit state.

• Intuitively, this task is equivalent to applying the Schur transform, projecting onto the isotypic subspaces of the unitary and symmetric groups indexed by the Young labels, and discarding of the permutation register.
• As such unitary Schur sampling is the natural task in processes such as quantum state tomography or spectrum estimation.
• We generalize this task to unitary mixed Schur sampling to account for the recently introduced mixed Schur-Weyl transform.