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

TL;DR

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, and shows that strict sequence recovery concentrates near the predicted coupling band.

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 $\widehat{P}_g(b)$ 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.

Figures (3)

• Figure 1: Sequential coupling protocol: after each hidden gate $g_i$, Eve couples a probe as $CR_x(\theta)$ and measures it mid-circuit, yielding a probe bit-string over depth $k$. Repeating $N$ times yields the histogram $\widehat{P}_g(b)$.
• Figure 2: Strict prefix accuracy cascade for prefix length $k=2$ over coupling $\theta$ and depolarizing noise $\lambda$, shown across shots $N$. This short-depth case serves as a sanity check for the depth-$2$ analysis in Sec. \ref{['sec:intractability-roots']}: accuracy collapses at trivial decoupling ($\theta\approx 0,2\pi$) and under strong noise, and otherwise saturates rapidly with $N$ when distinguishability is present. Dashed lines indicate $\theta^*(2)$ from eq. \ref{['eq:theta-star']} and its mirror $2\pi-\theta^*(2)$.
• Figure 3: Strict prefix accuracy cascade for prefix length $k=7$ over coupling $\theta$ and depolarizing noise $\lambda$, shown across increasing shots $N$. The coupling ridge (indicated by the vertical dashed lines) is the high-accuracy strength in $\theta$, which tracks the dashed predictor $\theta^*(7)$ from eq. \ref{['eq:theta-star']} (and the mirrored line $2\pi-\theta^*(7)$, since the proxy is symmetric under $\theta\mapsto 2\pi-\theta$). Noise (y-axis) lowers accuracy and narrows the viable coupling strength as $\lambda$ increases, while finite-shot effects (indicated by $N$) represent information density improvements in Eve's data.