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Abstract

We study a sequential coherent side-channel model in which an adversarial probe qubit interacts with a target qubit during a hidden gate sequence. Repeating the same hidden sequence for N shots yields an empirical full-correlation record: the joint histogram P_g (b) over probe bit-strings b {0, 1}[k], which is a sufficient statistic for classical post-processing under identically and independently distributed (i.i.d.) shots but grows exponentially with circuit depth. We first describe this sequential probe framework in a coupling- and measurement-agnostic form, emphasizing the scaling of the observation space and why exact analytic distinguishability becomes intractable with circuit depth. We then specialize to a representative instantiation (a controlled-rotation probe coupling with fixed projective readout and a commuting Rx gate alphabet) where we (i) derive a depth-dependent leakage envelope whose maximizer predicts a coupling band as a function of depth if the measurement data is reduced to marginal statistics, and (ii) provide an operational decoder, via machine learning, a single parameter-conditioned map from P_g to Alice's per-step gate labels, generalizing across coupling and noise settings without retraining.