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

The gist The study investigates a sequential coherent side-channel model where an adversarial probe qubit interacts with a target qubit during hidden gate sequences, yielding an empirical full-correlation record that serves as a sufficient statistic for classical post-processing under i.i.d. shots but scales exponentially with circuit depth

General Sequential Probe Model

The paper formalizes a sequential coupling-and-measurement threat model where Alice executes an unknown length-k gate string on a qubit A, and Eve repeatedly couples a probe qubit E after each gate and performs mid-circuit measurements on E The resulting record is history-dependent because statistics at step t depend on the prior gate context through the evolving joint state A typical execution yields a probe bitstring b = (y1,..., yk) ∈ 0, 1 k drawn from the underlying probability distribution Pg(b) induced by (V, My), where V is a fixed two-qubit unitary and My is a dichotomic measurement The structure of the scenario is summarized in Figure 1, showing Eve coupling a probe through the gate V (θ) and measuring it mid-circuit

Observation Model and Intractability

The key methodological choice is allowing Eve to retain the full correlation structure over time, yielding an empirical joint histogram Pbg(b) over probe bit-strings b ∈ 0, 1 k This full-correlation record is the natural sufficient statistic for classical post-processing under i.i.d. shots but it is high-dimensional (2 k) The paper analyzes both data reduced to its per-time-step marginals and the full data with explicit representational and inductive-bias choices for learning Exact analytic distinguishability becomes intractable with circuit depth The object P¯t,a is an expectation over all length-k continuations consistent with the sequence u, and its exact evaluation scales exponentially in both depth and outcome dimension

Analytic Predictor for Marginal Statistics

For a controlled-rotation probe coupling with fixed projective readout and a commuting Rx gate alphabet, the authors derive a depth-dependent leakage envelope and a closed-form maximum of the envelope, θ∗ (k) in Eq. (23) for the coupling strength that maximizes distinguishability under repeated interactions This leakage envelope highlights where coupling/noise are neither so small that Eve sees nothing, nor so large that the side-channel becomes either trivial or washed out, yielding maximal learnability in practice The analytic proxy for the gate-conditioned spread in a canonical per-step event probability is given by ∆pt(θ; G) ≈ αG sin2 θ / 2 cost−1 θ / 2 Maximizing ft(θ) = sin2 (θ/2) cost−1 (θ/2) yields the optimal coupling strength θ∗ (t) = 2 arcsin r squared t + 1, which shifts to smaller coupling with depth

Amortized Decoding from Full Correlations

The paper introduces supervised sequence decoders (neural models) trained to predict gates from Eve’s observed statistics, evaluating learnability via information metrics and trained decoders that attempt gate-by-gate reconstruction The Hist-TCN architecture converts the vectorized full-correlation histogram x into a length-k feature sequence X, which is processed by a dilated temporal convolutional network (TCN) over the step index t with a per-step regression head This yields an output aˆt interpreted as a normalized rotation angle φˆt/π at step t, from which gate labels are obtained by nearest-neighbor decoding The decoder is trained across a grid of (θ, λ) so that inference is amortized across physical regimes without retraining per grid point

Learning from Marginal Distribution and Full Correlation Data

When Eve uses only marginal statistics, the accuracy of guessing gate t correctly drops with depth and noise because successive probe interactions contract the Alice components that carry gate-dependent information into later marginal readouts Conversely, retaining the full joint histogram dramatically improves strict recovery, as shown in Fig. 3 (b), where increasing N from 64 to 4096 expands a broad high-accuracy region over θ The results support the view that coherent crosstalk should be analyzed as an inference channel over full measurement histories, with marginal envelopes serving as useful but incomplete interpretive guides

Conclusion

Operationally, Eve’s side-channel yields gate-level information in a strongly parameter-dependent way With only marginal statistics, Eve can recover early gates accurately in favorable coupling and low-noise regimes, but per-position accuracy degrades with depth and strict sequence recovery remains limited Retaining the full-correlation histogram changes the picture substantially, producing a much broader and higher-accuracy strict recovery region than marginal inputs Thus, the side-channel risk depends jointly on coupling strength, noise, shot count, depth, and the representation of Eve’s measurement record

How it works

** The sequential probe framework involves Alice applying a gate U(gt) on qubit A, followed by interaction with Eve's probe E via a fixed unitary V(θ), and then a dichotomic measurement My This process is interleaved with noise modeled by a CPTP map Nλ The resulting record is the empirical joint histogram Pbg(b) over probe bit-strings b ∈ 0, 1 k This correlation-rich record is the natural sufficient statistic for classical post-processing under i.i.d. shots **

Key Findings

1 The analysis distinguishes between two discrimination problems: position recovery, which asks whether the three mixture laws are distinguishable, and strict sequence recovery, which asks whether the full family of laws is distinguishable

2 Analytic distinguishability becomes intractable because evaluating exact equations scales exponentially in both depth and outcome dimension

3 The marginal statistics only contain gate information that becomes increasingly fragile with depth and noise, as successive probe interactions contract the Alice components that carry gate-dependent information into later marginal readouts

4 Retaining the full joint histogram dramatically improves strict recovery, converting the same underlying probe record into a much stronger strict sequence decoder input even at a small number of shots

Related Work and Positioning

The work positions itself by studying an estimator-independent constraint on operational learnability for sequence recovery when an adversary retains the full temporal correlation structure Pbg(b) The novelty lies in treating the classical channel generated by a coherent, history-dependent quantum interaction rather than conventional power, timing, or electromagnetic traces Unlike existing works that focus on classical telemetry or hardware-level crosstalk, this paper studies estimator-independent constraints and operational learnability for sequence recovery when an adversary retains the full temporal correlation structure Pbg(b)

Experimental Setup

The simulation uses a typical configuration with a gate alphabet G = 0, 1/8, 1/2, 1 acting on a target qubit A and depth k = 7 The coupling grid θ ∈ [0, 2π] and noise grid λ mapped to a depolarizing rate are varied Training data is generated by simulating the joint Alice–Eve dynamics for random gate sequences g, producing either marginal finite-shot statistics or full finite-shot histograms over Eve bitstrings Evaluation metrics include per-position gate accuracy and strict depth-7 sequence accuracy

References

[1] P. C.

1. textbfWeighting Probe Data by Correlation Structure: Hist-TCN Integration. The improved system will ingest full-correlation histogram data (Pbg) directly into a dilated temporal convolutional network (TCN) to produce gate labels, leading to a single, two-parameter-conditioned decoder fω: Pbg, θ, λ 7→ gˆ1:k instead of relying on marginal statistics alone.

2. textbfAmortized Decoder Training Across Physical Regimes. The system will be trained over a grid of coupling strength (θ) and noise scale (λ) to create a single decoder that interpolates across the full grid rather than retraining a model per coupling/noise setting, allowing for operational learnability without per-grid point retraining.

3. textbfQuantifying Information-Theoretic Limits. The system will utilize Total variation and KL divergence to establish estimator-independent limits, ensuring that the inference channel is analyzed based on whether two gate sequences induce nearly identical observation laws, thereby providing a rigorous, decoder-independent measure of distinguishability.