Hardware-Agnostic Modeling of Quantum Side-Channel Leakage via Conditional Dynamics and Learning from Full Correlation Data

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 widehat{P}_g(b) over probe bit-strings bin\{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 $R_x$ gate alphabet where we (i) derive a depth-dependent leakage envelope whose maximizer predicts a "Goldilocks" coupling band as a function of depth, and (ii) provide an operational decoder, via machine learning, a single parameter-conditioned map from widehat{P}_g to Alice's per-step gate labels, generalizing across coupling and noise settings without retraining. Experiments over broad coupling and noise grids show that strict sequence recovery concentrates near the predicted coupling band and degrades predictably under decoherence and finite-shot estimation.