Variational quantum circuits (VQCs) have been proposed as promising models for nonlinear learning tasks in the noisy intermediate-scale quantum (NISQ) era. This study presents a hybrid classical-quantum framework for nonlinear time-series forecasting that integrates causally-filtered Hidden Markov regime detection with multiple VQC architectures. Four circuit architectures are defined as an explicit [Formula: see text] factorial design that isolates circuit depth from entanglement topology (shallow: depth 1/linear; ring: depth 1/ring; deep: depth 4/linear; heavy: depth 4/ring), each with reported qubit count, parameter count and gate count, and are evaluated under an analytic depolarizing/readout noise model. Three nonlinear classical baselines (support-vector regression, gradient boosting, and a multilayer perceptron) are compared alongside a linear autoregressive (AR) baseline, and the Hidden Markov Model (HMM) regime feature is fit only on training data and propagated causally into the test set to remove look-ahead leakage. Across 20 Monte Carlo replicates, we find that every VQC architecture is outperformed by every classical baseline, including plain linear AR (paired t-tests, all [Formula: see text]; Cohen’s d up to 4.8). Within the VQC family, circuit depth - not shallowness - is associated with better performance (deep/heavy [Formula: see text] shallow/ring), while entanglement topology alone has little independent effect once depth is controlled. An ablation removing the HMM-derived feature shows it contributes negligibly to forecast accuracy in this setting. Under this confound-controlled design, we find no evidence of a forecasting advantage for the tested VQC architectures over classical linear or nonlinear baselines, which we report as a cautionary result for claims of near-term quantum advantage in forecasting.
In many natural and engineered systems, unknown quantum channels act on a subsystem that cannot be directly controlled and measured, but is instead learned through a controllable subsystem that weakly interacts with it. We study quantum channel discrimination (QCD) under these restrictions, which we call hidden system QCD. We find sequential protocols achieve perfect discrimination and saturate the Heisenberg limit. In contrast, depth-1 parallel and multishot protocols cannot solve hidden system QCD. This suggests sequential protocols are superior in experimentally realistic situations.
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 \emph{full-correlation record}: the joint histogram Pg(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 Pg to Alice's per-step gate labels, generalizing across coupling and noise settings without retraining.