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 $b\in\{0,1\}^k$, 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.

翻译：我们研究一种顺序相干侧信道模型，其中对抗性探测量子比特在隐藏门序列期间与目标量子比特相互作用。对同一隐藏序列重复进行$N$次实验可获得经验全关联记录：探测比特串$b\in\{0,1\}^k$上的联合直方图$\widehat{P}_g(b)$，该统计量在独立同分布实验条件下是经典后处理的充分统计量，但其维度随电路深度呈指数增长。我们首先以耦合与测量无关的形式描述该顺序探测框架，重点分析观测空间的标度特性，并阐释精确解析可区分性为何随电路深度变得难以处理。随后我们聚焦于典型实例化方案（采用受控旋转探测耦合与固定投影读取，配合对易的$R_x$门集合），在此框架下我们（i）推导出深度相关的泄漏包络函数，其最大值预测了随深度变化的"适中"耦合带；（ii）通过机器学习提供可操作解码器：构建从$\widehat{P}_g$到Alice每步门标签的单参数条件映射，该映射能泛化至不同耦合与噪声环境而无需重新训练。在宽范围耦合与噪声网格上的实验表明，严格序列恢复集中在预测耦合带附近，且在退相干与有限次实验估计条件下呈现可预测的性能退化。