Abstract Circuit topology—the pattern of two-qubit entangling gates—determines noise propagation paths in variational quantum circuits (VQCs). Using the normalized rank-displacement diagnostic D(ε) across 26 experiment configurations spanning five topologies and five CPTP noise types, plus 135 IBM ibm fez hardware circuits, we show that topology accounts for 48.9% of explainable D(0.05) variance under ω 2 decomposition (Cohen’s d = 2.66 between ring and star; permutation p = 7.1×10−12). The ranking hierarchy (star>grid≈linear>ring, where lower D indicates better ranking preservation) is invariant across the four nondepolarizing noise types (Kendall τ = 1.0 over four ranked lists, p = 0.017 by exact permutation test) but collapses under depolarizing noise. On IBM hardware, the qualitative ranking survives transpilation, though effect magnitude attenuates ∼ 10×. A Heisenberg-picture analytical model qualitatively predicts the τ (n, L) ranking from circuit structure alone (r = 0.645, p = 0.013; R2 = 0.35), providing a theoretical basis for topology-dependent noise sensitivity. We find that classical ML systems (MLP, XGBoost) achieve comparable D(0.05) values, showing that the τ -D functional form is consistent across classical and quantum computation, suggesting a shared noise-response structure. Practical design heuristics are provided
Abstract Circuit topology—the pattern of two-qubit entangling gates—determines noise propagation paths in variational quantum circuits (VQCs). Using the normalized rank-displacement diagnostic D(ε) across 26 experiment configurations spanning five topologies and five CPTP noise types, plus 135 IBM ibm fez hardware circuits, we show that topology accounts for 48.9% of explainable D(0.05) variance under ω 2 decomposition (Cohen’s d = 2.66 between ring and star; permutation p = 7.1×10−12). The ranking hierarchy (star>grid≈linear>ring, where lower D indicates better ranking preservation) is invariant across the four nondepolarizing noise types (Kendall τ = 1.0 over four ranked lists, p = 0.017 by exact permutation test) but collapses under depolarizing noise. On IBM hardware, the qualitative ranking survives transpilation, though effect magnitude attenuates ∼ 10×. A Heisenberg-picture analytical model qualitatively predicts the τ (n, L) ranking from circuit structure alone (r = 0.645, p = 0.013; R2 = 0.35), providing a theoretical basis for topology-dependent noise sensitivity. We find that classical ML systems (MLP, XGBoost) achieve comparable D(0.05) values, showing that the τ -D functional form is consistent across classical and quantum computation, suggesting a shared noise-response structure. Practical design heuristics are provided
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.
Abstract Variational quantum circuits (VQCs) are a central paradigm for quantum machine learning, yet their practical deployment is limited by poor trainability, sensitivity to noise, and the lack of input-adaptive parameterization. In this work, we introduce TensorDual-VQC, a tensor-structured operator-learning framework that models variational quantum computation as an input-conditioned mapping that jointly governs quantum state encoding and circuit parameter generation. The proposed approach employs a shared low-rank Tensor-Train (TT) backbone with dual output heads, inducing a coupled representation that constrains the hypothesis space to a structured low-dimensional manifold. Under standard smoothness and low-rank assumptions, we show that the approximation behavior is governed by the TT-rank rather than the ambient parameter dimension, and that shared latent structure can enable more compact representations than decoupled parameterizations in structured regimes. We further analyze trainability through the induced Neural Tangent Kernel, showing that the multi-branch gradient structure enriches the span of gradient features and can improve kernel conditioning under mild non-degeneracy conditions. In addition, we relate robustness to quantum noise to the intrinsic effective dimension of the tensor parameter manifold, providing a first-order scaling characterization of output variance under stochastic perturbations. We evaluate TensorDual-VQC on quantum dot classification, quantum simulation, and combinatorial optimization tasks under both simulated noise and real quantum hardware. Across all settings, the proposed framework achieves higher accuracy, faster convergence, and greater stability than standard VQCs and hypernetwork-based baselines when quantum resources are matched. These results suggest that structured operator learning with shared low-rank representations provides a principled and practical approach to scalable, noise-resilient quantum learning in the NISQ regime.
Variational quantum circuits (VQCs) are a leading approach to quantum machine learning on near-term devices, yet it remains unclear which circuit architecture yields the best accuracy-parameter trade-off on classical tabular data. We present a systematic empirical comparison of four VQC families -- multi-layer fully-connected (FC-VQC), residual (ResNet-VQC), hybrid quantum-classical transformer (QT), and fully quantum transformer (FQT) -- across five regression and classification benchmarks. Our key findings are: \textbf{(i)}~FC-VQCs achieve 90-96\% of the R2 of attention-based VQCs while using 40-50\% fewer parameters, and consistently outperform equal-capacity MLPs (mean R2=0.829 vs.\ MLP720's 0.753 on Boston Housing, 3-seed average); \textbf{(ii)}~FC-VQC's Type~4 inter-block connectivity provides partial cross-token mixing that approximates the role of attention -- explicit quantum self-attention yields only marginal gains on most datasets while significantly increasing parameter count; \textbf{(iii)}~expressibility saturates at circuit depth~≈3, explaining why shallow VQCs already cover the Hilbert space effectively; \textbf{(iv)}~LayerNorm on the fully quantum transformer improves classification accuracy, suggesting normalization is important when all operations are quantum; \textbf{(v)}~in our noise study on Boston Housing, FQT degrades gracefully under depolarizing noise while QT collapses. All results are validated across three random seeds. These findings provide practical architectural guidance for deploying VQCs on near-term quantum hardware.
Variational quantum circuits (VQCs) -- parameterised quantum circuits trained by classical optimisation of a quantum cost function -- underpin most near-term quantum machine learning, yet systematic guidance on design choices (ansatz structure, encoding, optimiser, gradient estimation) and their interaction with specific ML task requirements remains fragmented. This paper proposes the Variational Quantum Circuit Design Space (VQCDS) framework, characterising 48 VQC configurations (3 ansatz families x 3 encoding strategies x 4 optimisers) across 12 ML benchmark tasks -- regression, classification, generative modelling, and reinforcement learning -- on IBM Quantum Eagle and IonQ Forte. VQCDS finds: LSAD-structured ansatz (Silva et al., 2024) outperforms hardware-efficient ansatz on 10 of 12 tasks (+8.0pp mean AUC, 2.1x lower variance); amplitude encoding outperforms angle encoding for p > 16 features (+12.4% R2); quantum natural gradient descent (QNG) reduces regression convergence by 48.4% vs. Adam; and task-ansatz alignment via mutual information-guided entanglement provides +8.4pp AUC and +14.4% R2 across task types. VQCDS provides an evidence-based VQC design guide covering 48 configuration evaluations across 12 tasks.
Parameterized quantum circuits (PQCs) serve as a flexible framework for implementing various quantum algorithms, particularly in quantum machine learning and optimization tasks. However, their training is challenging due to both theoretical constraints and the inherent limitations in current quantum hardware. A judicious choice of PQC architecture is crucial for accurately representing the solution space, and for ensuring convergence in training. Several works introduced measures of expressibility and entangling capability for PQCs, and compared these metrics across different circuit structures, providing insights into how they impact performance and efficiency. However, these investigations often assumed idealized all-to-all qubit connectivity, which is not applicable to most hardware architectures. In practice, additional transpilation steps are performed prior to execution on hardware. The potential impact of better alignment of the PQC structure with the underlying native topology has in fact motivated introduction of hardware-efficient ansatze. In our work, we extend the methodology of measuring PQC efficiency to include more realistic quantum hardware models. We examine several popular hardware topologies, including hexagonal, square, circular, linear, and all-to-all. We evaluate the expressibility and entangling capability of circuits with different connectivity types and various arrangements of rotation gates. We assess the training performance of these PQC types in the context of a variational compiling procedure and identify the preferred gate patterns. This approach provides a more realistic evaluation of PQC performance, thus offering practical insights into their applications on existing quantum devices.
The selection of the right topology (ansatz) for a Variational Quantum Algorithm (VQA) is a complex task that usually involves deep knowledge of a particular problem. The importance of the selection is greater when we consider the current state of quantum hardware, particularly the noise associated with the complexity of Variational Quantum Circuits (VQCs) that implement VQAs. Here, a comparison is presented between two confronted approaches for solving the MaxCut problem: QAOA, which has a theoretical proof of convergence, and the automatic design proposal (QNAS), which relies on evolutionary algorithms (NSGA-II) to discover efficient circuits. The comparison was made across 490 graph instances from different graph topologies and sizes (n=4 to n=16), accounting for noise models such as depolarizing noise, gate errors, and readout noise. The results show that QAOA achieves an approximation ratio (rA) ≈1 on complete graphs at the cost of being almost 12 times more complex than QNAS in ideal conditions while approaching the random noise floor (rA≈0.5). QNAS was capable of finding circuits less complex while maintaining 69% of the fidelity at a cost of having an rA on the interval 0.7≤rA≤0.8. However, when the comparison is made across sparse graphs, performance is comparable, while QNAS is less complex.