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.
Journal reference
2026 IEEE International Conference on Quantum Control, Computing, and Learning (qCCL), 2026
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 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 "Goldilocks" coupling band as a function of depth, 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. 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.
We propose an explicit feature representation for supervised learning derived from short quantum dynamical trajectories composed of generally noncommuting operations, conceptually inspired by reservoir computing in that fixed, untrained dynamics are coupled to a simple trained readout. For each input, a fixed-depth circuit generates a sequence of prefix states, and features are formed by concatenating expectation values of a predetermined set of single- and two-qubit Pauli observables recorded across depths; multiple randomized but fixed circuit instantiations are aggregated to improve stability. A classicall2- regularized logistic-regression readout is trained on the resulting explicit features. To isolate representational effects, all methods are evaluated under identical preprocessing, repeated stratified holdout splits, and a unified model-selection protocol on a controlled subset of 11 tabular classification datasets, and are compared against classical baselines, including random Fourier features and second-degree polynomial expansions, as well as reservoir-style quantum baselines, with feature dimensionality matched when feasible. We further examine controlled variants targeting commutativity, entangling structure, history concatenation, and observable choice. A staged evaluation contrasts an idealized exact-expectation regime with finite-shot estimation under fixed and larger measurement budgets, thereby assessing sensitivity to sampling noise after re-tuning regularization. Beyond predictive performance, we analyze the induced geometry via empirical kernel spectra and effective-dimension diagnostics to contextualize the representations from a regularization-aware capacity perspective. Empirically, the strongest support concerns noncommutativity and history concatenation, whereas the contribution of the entangling block is mixed; the trajectory-based features induce empirically distinct kernel geometries, while finite-shot robustness remains strongly task-and configuration-dependent.
Physical computational devices leak side-channel information that may, and often does, reveal secret internal states. We present a general transformation that compiles any circuit into a circuit with the same functionality but resilience against well-defined classes of leakage. Our construction requires a small, stateless, and computation-independent leak-proof component that draws random elements from a fixed distribution. In essence, we reduce the problem of shielding arbitrarily complex circuits to the problem of shielding a single, simple component. Our approach is based on modeling the adversary as a powerful observer that inspects the device via a limited measurement apparatus. We allow the apparatus to access all the bits of the computation (except those inside the leak-proof component), and the amount of leaked information to grow unbounded over time. However, we assume that the apparatus is limited in the amount of output bits per iteration and the ability to decode certain linear encodings. While our results apply in general to such leakage classes, in particular, we obtain security against (a) constant-depth circuits leakage, where the leakage function is computed by an AC0 circuit (composed of NOT gates and unbounded fan-in AND and OR gates); (b) noisy leakage, where the leakage function reveals all the bits of the internal state of the circuit, but each bit is perturbed by independent binomial noise---i.e., flipped with some probability p. Namely, for some number p∈(0,1/2], each bit of the computation is flipped with probability p, and remains unchanged with probability 1−p.
Abstract Understanding how depth-dependent quantum evolution reshapes the geometric organization of data representations in quantum neural networks (QNNs) remains a central question in quantum machine learning. We study a hardware-efficient variational QNN—single-qubit R Y / R Z rotations per layer followed by a nearest-neighbor CNOT entangling ring, with circuit depth n ranging from 3 to 12 layers—and ask how its data representation geometry reorganizes as n increases. We report two findings across MNIST and BloodMNIST (15 independent random seeds for amplitude encoding: 5 MNIST, 10 BloodMNIST), with a BloodMNIST angle-encoding control (10 further seed-runs, 25 in total) used to check the second finding’s recurrence under a different encoding strategy. First, class separability shows power-law growth within the tested depth window, S ( n ) ∝ n b , where b = δ − ε captures the balance between inter-class expansion and intra-class diffusion; this description holds throughout the tested range for MNIST, whereas for BloodMNIST the growth is better described by a model approaching a depth-independent ceiling within the same range, with b retained as a common cross-dataset summary exponent. Second, we measure an auxiliary parameter-perturbation sensitivity statistic and find that its depth-trend exponent γ shows a dataset-level association with b via b ≈ k data ( 1 − | γ | ) . We explicitly tested whether this statistic tracks the textbook state-based quantum Fisher information by computing the latter rigorously for the same trained circuits, and found that it does not; we also find that the b – γ association is weak and not statistically significant within either dataset alone. We therefore report this coupling descriptively, as an open dataset-level pattern rather than an established physical law. To our knowledge, this is the first report of this depth-dependent separability-growth pattern in QNNs together with a practical, predictive heuristic for depth selection derived from it, using each dataset’s own functional form. In the tested depth regime ( n ∈ [ 3 , 12 ]
Classical shadows extract quantum properties from reusable classical records, typically obtained through randomized measurement settings. Although individual settings may be shallow, switching among them introduces control, calibration, and reconfiguration costs beyond conventional metrics. Here we construct, for every n, a single fixed shallow quantum analyzer whose Born outcomes replace externally sampled settings as labels for reusable shadow snapshots. The analyzer combines a freshly prepared n-qubit fiducial register A with parallel Bell readout of A and the unknown system S, producing one 2n-bit record per copy. The same circuit realizes a rank-one minimal informationally complete measurement: one fixed setting replaces the 3n local-Pauli settings conventionally used for complete reconstruction while retaining the minimum d2 outcomes with d=2n. Its Pauli-diagonal frame admits an analytic inverse. For fixed Hermitian observables, the Haar-averaged conditional variance has a dimension-independent coefficient, whereas the state-uniform coefficient is dimension dependent. For n≥3, worst-state Pauli variances remain bounded in the axial sector and scale as Θ(d) in the mixed sector. Fiducial preparation before system contact uses n−1 arbitrary two-qubit gates, no work qubits beyond A, and logarithmic depth under all-to-all connectivity. The unknown system undergoes one parallel system-ancilla entangling layer followed by local Hadamards and readout. Measurement-assisted preparation achieves O(1) adaptive quantum depth using n+1 extra qubits in the heralded route, or O(nlogn) auxiliaries for deterministic completion. These tradeoffs show that part of the measurement-setting randomness and control complexity normally supplied shot by shot can instead be compiled into a fixed reusable analyzer.
We introduce a method for designing smooth single-qubit control pulses that implement a desired gate while suppressing the effect of unknown static error sources to first order. Unlike dynamically corrected gate constructions that require prior knowledge of the noise model, the present approach is agnostic to the detailed form of the target-bath interaction. The method parametrizes the control propagator through an auxiliary matrix expansion over orthogonal basis functions and enforces decoupling through algebraic orthogonality and equal-norm constraints on the expansion coefficients. These conditions guarantee that the leading Magnus contribution of an arbitrary static interaction reduces to a term proportional to the identity on the target system, thereby cancelling first-order error effects independently of the microscopic origin of the noise. We further show that the same construction suppresses, to first order, mediated couplings between simultaneously controlled qubits when their interaction occurs through intermediate environmental degrees of freedom, yielding effective second-order decoupling of the induced inter-qubit interaction. By using a discrete cosine transform parametrization, the pulse-synthesis problem is cast into a numerically stable constrained optimization with a minimal number of free parameters. Numerical examples for Rz rotations and random single-qubit unitaries demonstrate smooth control fields that realize the target gates while remaining robust against arbitrary static single-qubit noise and mediated multi-qubit couplings. These results provide a hardware-friendly route toward noise-agnostic dynamically corrected single-qubit gates.
The design of high-fidelity quantum gates is difficult because it requires the optimization of two competing effects, namely maximizing gate speed and minimizing leakage out of the qubit subspace. We propose a deep reinforcement learning algorithm that uses two agents to address the speed and leakage challenges simultaneously. The first agent constructs the qubit in-phase control pulse using a policy learned from rewards that compensate short gate times. The rewards are obtained at intermediate time steps throughout the construction of a full-length pulse, allowing the agent to explore the landscape of shorter pulses. The second agent determines an out-of-phase pulse to target leakage. Both agents are trained on real-time data from noisy hardware, thus providing model-free gate design that adapts to unpredictable hardware noise. To reduce the effect of measurement classification errors, the agents are trained directly on the readout signal from probing the qubit. We present proof-of-concept experiments by designing X and square root of X gates of various durations on IBM hardware. After just 200 training iterations, our algorithm is able to construct novel control pulses up to two times faster than the default IBM gates, while matching their performance in terms of state fidelity and leakage rate. As the length of our custom control pulses increases, they begin to outperform the default gates. Improvements to the speed and fidelity of gate operations open the way for higher circuit depth in quantum simulation, quantum chemistry and other algorithms on near-term and future quantum devices.