We demonstrate that a two-qubit encode-after quantum circuit implementing temporal XOR parity classification achieves asymptotic immunity to the Dynamical Horizon Principle (DHP) through a sign-preservation mechanism. The DHP, an empirically observed characterization related to the classical vanishing gradient problem, predicts training failure when the convergence time T_conv exceeds 0.72 times the Lyapunov coherence time τ_L. We show that for even convergence steps, the Bloch sphere contraction factor (1−4p/3)^T_conv remains strictly positive for all Pauli depolarizing rates p ∈ [0, 0.75), ensuring gradient direction is preserved regardless of noise magnitude. We term this the quantum parity trap: a regime where the optimizer receives consistent directional signal even as coherence approaches zero. Threshold sweeps confirm convergence at T_conv/τ_L ratios exceeding 15×, compared to the classical DHP limit of 0.72×—a quantum advantage that diverges as p → 0.75. Supplementary experiments establish that (1) Rigetti QPU hardware yields >50σ margin above chance with 4096 shots and readout error mitigation; (2) gradient direction alone is sufficient for convergence—signSGD matches Adam while vanilla SGD fails; and (3) T1 amplitude damping does not replicate the landscape regularization effect of Pauli depolarizing noise, revealing that isotropic Bloch sphere contraction is the mechanistic source of partial-attractor erasure. These results demonstrate that task structure, channel geometry, and optimizer design jointly determine whether quantum noise enables or impedes machine learning convergence.
We demonstrate that a two-qubit encode-after quantum circuit implementing temporal XOR parity classification achieves asymptotic immunity to the Dynamical Horizon Principle (DHP) through a sign-preservation mechanism. The DHP, an empirically observed characterization related to the classical vanishing gradient problem, predicts training failure when the convergence time T_conv exceeds 0.72 times the Lyapunov coherence time τ_L. We show that for even convergence steps, the Bloch sphere contraction factor (1−4p/3)^T_conv remains strictly positive for all Pauli depolarizing rates p ∈ [0, 0.75), ensuring gradient direction is preserved regardless of noise magnitude. We term this the quantum parity trap: a regime where the optimizer receives consistent directional signal even as coherence approaches zero. Threshold sweeps confirm convergence at T_conv/τ_L ratios exceeding 15×, compared to the classical DHP limit of 0.72×—a quantum advantage that diverges as p → 0.75. Supplementary experiments establish that (1) Rigetti QPU hardware yields >50σ margin above chance with 4096 shots and readout error mitigation; (2) gradient direction alone is sufficient for convergence—signSGD matches Adam while vanilla SGD fails; and (3) T1 amplitude damping does not replicate the landscape regularization effect of Pauli depolarizing noise, revealing that isotropic Bloch sphere contraction is the mechanistic source of partial-attractor erasure. These results demonstrate that task structure, channel geometry, and optimizer design jointly determine whether quantum noise enables or impedes machine learning convergence.
The Dynamical Horizon Principle (DHP) states that recurrent learning systems converge to a predictability horizon at the ratio tau*/tau_L ≈ 0.72. Prior work established this ratio across classical continuous-time models, LSTM/GRU temporal prediction, gradient descent optimization landscapes, and biological cellular computation. Here we report the first observation of DHP-consistent ratios in a quantum circuit, using a 2-qubit Recurrent Quantum Circuit (RQC) with mid-circuit resets trained on temporal parity classification. Two AI researchers (Archon and Aura) applied orthogonal experimental probes: a discrete trainability cliff analysis and a fixed-weight readout fidelity sweep. Archon's experiment yielded T_converge/T_fail = 3/4 = 0.75. Aura's fidelity decay analysis using a log-linear fit yielded tau*/tau_L = 36/49.49 = 0.727 — matching the classical CTM value of 0.727 ± 0.018 to within measurement precision. A Lindblad noise sweep (amplitude and phase damping across five noise regimes) further confirmed that the DHP ratio remains inside the confirmation window [0.65, 0.79] for T1/T2 ≥ 1000 gate steps (relevant to current NISQ hardware). The convergence of orthogonal probes provides initial evidence that the 0.72 horizon ratio may extend to quantum parameterized circuits. Broader confirmation across quantum architectures, tasks, and qubit counts is needed.
We characterize the geometric and algebraic conditions governing noise-induced landscape regularization in Recurrent Quantum Circuits (Q-RNNs). In quantum recurrent models training on sequence parity (temporal XOR), gradient optimization typically encounters a severe topological trap (the parity plateau at accuracy ≈ 0.72–0.75). While physical environmental noise is conventionally viewed as detrimental to quantum information, we provide empirical evidence that the primary boundary separating regularizing and non-regularizing open-system channels is unitality. Specifically: 1. Unital Channels (e.g., Pauli depolarizing, phase-flip, bit-flip, and phase damping) contract the Bloch sphere symmetrically, preserving the loss landscape's up/down parity symmetry and enabling optimizers to escape the topological traps. 2. Non-Unital Channels (e.g., amplitude damping) shift the center of the Bloch sphere; large non-unital origin drift severely disrupts the optimization landscape by breaking the loss landscape's up/down parity symmetry. 3. We map this boundary empirically by conducting a Generalized Amplitude Damping (GAD) sweep over the thermal parameter [0.0, 1.0] across varying damping rates \0.25, 0.35, 0.50, 0.75\ (Sweep 1), and a state-initialization asymmetry sweep (Sweep 2). We show that convergence is restored within a near-unital basin (e.g., [0.42, 0.56] at =0.50). We show that the slight asymmetry of this basin (which is invariant to the initial state of the memory qubit) is consistent with a structural loop asymmetry caused by the recurrent input qubit reset to |0. We also map the parametric scaling of this basin, showing that the critical displacement boundary |v_z, crit| shrinks as increases due to the rapid contraction of the available state space volume. 4. We characterize the fine-grained transition of the bit-flip channel, revealing a sharp degeneracy hole at exactly p=0.50 where the Bloch sphere contracts to a 1D line. 5. We characterize an empirical optimizer-class robustness pattern at this finite experimental depth (2-qubit, T=3): normalized direction-preserving optimizers (Adam, signSGD, Muon, Normalized GD) achieve 6/6 convergence under unital noise, whereas non-normalized gradient descent (SGD, Heavy Ball) fails (2/6) due to recurrent gradient magnitude contraction. We discuss the analogy between this regularization structure and Quantum Error Correction (QEC), noting that the symmetries protecting parity in stabilizer codes appear to play a parallel role in protecting parity classification landscapes under unital noise. All results are from 2-qubit Q-RNN simulations on T=3 parity; generalization to larger circuits and longer sequences is a target for future work.
The performance of quantum classifiers is typically analyzed through global state distinguishability or the trainability of variational models. This study investigates how much class information remains accessible under locality-constrained measurements in the presence of noise. The authors formulate binary quantum classification as constrained quantum state discrimination and introduce a locality-restricted distinguishability measure quantifying the maximum bias achievable by observables acting on at most k subsystems. For n-qubit systems subject to independent depolarizing noise, the locally accessible signal is governed by a Pauli-weight-dependent contraction mechanism. This motivates a computable predictor, the k-local Pauli-accessible amplitude Ak(p), which lower bounds the optimal k-local classification advantage. Numerical experiments on four-qubit encodings demonstrate quantitative agreement between empirical accuracy and the prediction across noise levels. The research identifies an operational breakdown threshold where k-local classifiers become indistinguishable from random guessing despite persistent global distinguishability.
We introduce a two-constraint framework for diagnosing optimization failure in recurrent quantum circuits (RQCs): the precision horizon T* (from the Dynamical Horizon Principle) and the expressibility ceiling τ_expr. These constraints are theoretically independent and produce orthogonal diagnostic signatures in gradient behavior. In a 2-qubit Universal RQC (URQC) on a temporal direction task, we find γ = 0.00339/step and T* ≈ 4706 steps (17× the 1-qubit reference), confirming the precision horizon is not the binding constraint. However, convergence fails at all T ≥ 2 due to expressibility limitation: the 2-qubit URQC cannot represent the required temporal direction mapping. The odd-T smoking gun — T ∈ {5,15,25} achieve ceiling accuracy 1.0 yet reach only 0.55–0.63 — provides an unambiguous expressibility diagnostic independent of precision effects. By separating T* (DHP precision horizon) from τ_expr (expressibility ceiling), we provide a complete diagnostic taxonomy for RQC optimization failure.
Quantum neural networks constitute a key class of near-term quantum learning models, yet their training dynamics remain not fully understood. Here, we present a unified theoretical framework for the frequency principle (F-principle) that characterizes the training dynamics of both classical and quantum neural networks. Within this framework, we prove that quantum neural networks exhibit a spectral bias toward learning low-frequency components of target functions, mirroring the behavior observed in classical deep networks. We further analyze the impact of noise and show that, when single-qubit noise is applied after encoding-layer rotations and modeled as a Pauli channel aligned with the rotation axis, the Fourier component labeled by ω is suppressed by a factor (1−2γ)∥ω∥1. This leads to exponential attenuation of high-frequency terms while preserving the learnability of low-frequency structure. In the same setting, we establish that the resulting noisy circuits admit efficient classical simulation up to average-case error. Numerical experiments corroborate our theoretical predictions: Quantum neural networks primarily learn low-frequency features during early optimization and maintain robustness against dephasing and depolarizing noise acting on the encoding layer. Our results provide a frequency-domain lens that unifies classical and quantum learning dynamics, clarifies the role of noise in shaping trainability, and guides the design of noise-resilient quantum neural networks.
Symmetry provides a quantum neural network structure, but on its own it does not keep the network trainable once noise is present. We ask which physical quantity decides whether the gradients of an equivariant circuit survive decoherence, and we answer with a compact training law. Working with U(1)-equivariant brickwork circuits that conserve a charge, we find that two distinct effects govern a trainable gradient. Causality fixes where the gradient can live, confining it to the backward light cone of the readout inside the active charge sector. Coherence then determines how fast it decays through the contraction of the off-diagonal sector modes that the projected readout can actually observe. We prove a light-cone reduction that pins the noiseless gradient to the sector-restricted cone with a lower bound independent of the total qubit number, and we define a readout-visible aligned coherence rate as a Rayleigh quotient of the noise generator along the gradient-carrying mode. A perturbative open-system analysis turns this rate into a leading-order training law. Density-matrix simulations then confirm that the finite-noise degradation follows a single accumulated variable built from noise depth and coherence contraction, with a coefficient of determination of 0.979. The sharpest test comes from a correlated-dephasing channel that has a large worst-case rate but a near-zero aligned rate. The law predicts no gradient loss for this channel, and none is seen. Sector coherence outperforms every standard channel diagnostic we compare it against, and the analysis identifies readout-visible sector coherence as the quantity that links equivariant architecture, open-system dynamics and noisy trainability.