COEP Technological University

A quantum recurrent map keeps its memory as a density matrix and implements the forget gate as an input-dependent replacement channel; its output depends on any single input as a trigonometric polynomial whose frequency set is fixed by the encoding, and under a constant gate the nonzero-frequency coefficients decay with that input's age at a rate set by the gate. The sensitivity bound for each input contains the encoding scales, layer count, observable norm and gate values but no trainable unitaries, and it sums uniformly in sequence length; the same weights yield an unbiased estimator built only from reset-free unitary circuits, with expected depth at most (1−fmax)⁻¹.

A quantum recurrent map keeps its memory as a density matrix and implements the forget gate as an input-dependent replacement channel; its output depends on any single input as a trigonometric polynomial whose frequency set is fixed by the encoding, and under a constant gate the nonzero-frequency coefficients decay with that input's age at a rate set by the gate. The sensitivity bound for each input contains the encoding scales, layer count, observable norm and gate values but no trainable unitaries, and it sums uniformly in sequence length; the same weights yield an unbiased estimator built only from reset-free unitary circuits, with expected depth at most (1−fmax)⁻¹.