Physical channels, not a quantum GRU

Quantum reservoir computing provides a precedent for input injection, finite quantum dynamics, and readout-based sequential processing FujiiNakajima. The construction below is not a re-interpretation of the GRU or a transfer of that article's numerical comparisons.

Let ${\mathcal{H}}_Q$ be finite dimensional and let $s$ be a fixed structural sector during one sequence. Declare CPTP maps

For example, an input-dependent finite GKSL generator gives ${\Lambda }_{x,s}=e^{\Delta t{\mathcal{L}}_{x,s}}$ GKS,Lindblad. A graph-based choice can use the coherent and jump-rate architecture of quantum stochastic walks Whitfield,CheungPsy03A.

An alternative declared injection on a composite carrier $Q=A\otimes R$ is

followed by a fixed finite channel. This injection is CPTP. It does not apply a nonlinear activation to an unknown state.

Complete record probabilities require instruments. Ideal expectation-valued features also require a measurement or repeated-preparation protocol; they are not free noiseless records from one unknown state Watrous,CheungPsy02A.