Three-qubit residual and its reading instrument

A direct finite implementation uses one fresh probe qubit $Q$. For fixed $k,j$, apply the real rotation whose first column is

Control it diagonally by $j$, then destructively read $Q$. There are only $32$ specified controlled rotations. They are admitted finite gates of this model; no hardware compilation or gate-synthesis error is asserted.

The instrument is defined on coherent superpositions of $|j⟩$ as well. Such superpositions are probe states. They are not identified with superpositions of the continuum classical field configurations.