Variational Quantum Circuits

Quantum neural networks (QNNs) are typically built from variational quantum circuits (VQCs), which are limited by local measurements. Adaptive non-local observables (ANO) address this by jointly optimizing circuit parameters and multi-qubit measurements. However, existing ANO-based VQCs learn only a single static observable that remains invariant across all inputs. We propose QFWP-ANO, a novel architecture which employs a classical hypernetwork to dynamically program VQC parameters and/or non-local observables conditioned on each input. On multivariate time-series forecasting across four ETT datasets, QFWP-ANO achieves the lowest MSE in 16 of 20 settings and second-lowest in the remaining four, surpassing ANO-based and other strong baselines. On reinforcement learning tasks, QFWP-ANO consistently surpasses ANO-VQCs. Our results establish input-conditioned ANO as an effective approach for enhancing QNNs.

Quantum Reinforcement Learning (QRL) represents policies as variational quantum circuits (VQCs), making it attractive for combinatorial optimization such as the Capacitated Vehicle Routing Problem (CVRP). On noisy intermediate-scale quantum (NISQ) hardware, however, decoherence degrades fidelity and destabilizes learning, and conventional error mitigation is applied statically without regard to the learning context. We introduce Adaptive Policy-Guided Error Mitigation (APGEM), a controller that selects among Zero-Noise Extrapolation (ZNE), Probabilistic Error Cancellation (PEC), Clifford Data Regression (CDR), and Readout Error Mitigation (REM) online, driven by a fidelity, entropy, and cost aware utility function and an epsilon-greedy rule over temporal-difference Q-scores. We evaluate on a realistic urban-logistics testbed, a Delhi-based CVRP over real landmarks with geodesic inter-node costs, exercised across five noise families and four severity levels. On this instance, the QRL agent outperforms constructive heuristics and approaches metaheuristics, while mitigation restores approximation ratios from 0.84-0.87 to 0.92-0.94 under high noise. The controller shifts from a CDR-dominated regime under short training horizons to a balanced deployment across all four techniques under longer horizons, indicating genuine regime-dependent selection. These preliminary results position adaptive, learning-aware mitigation as a practical route to noise-resilient QRL.

Variational quantum circuits (VQCs) are parameterized quantum circuits optimized classically. We propose a hybrid variational quantum circuit (HVQC) extending VQCs with a classical affine post-measurement layer, enabling vector-valued regression without the linear overhead of independent scalar circuits. Theoretically, we show that elementary one-and two-qubit circuits can approximate quadratic functions and products via data re-uploading and entanglement, providing the foundations of the full architecture. Experimentally, on two synthetic image reconstruction datasets and the Friedman1 benchmark (40,568 test samples), our HVQC matches Gaussian Process Regression and outperforms XGBoost and Random Forest. An ablation study confirms that both quantum and classical components are essential, and results highlight the central role of the feature map in hybrid quantum-classical models.

Quantum generative adversarial networks (QGANs) have attracted increasing attention for image generation using parameterized quantum circuits. Existing amplitude-based approaches face two key limitations: pixel locations are typically encoded by computational-basis indices or address qubits, causing quantum resources to grow with image resolution; meanwhile, jointly decoding many pixels from normalized quantum states introduces probability competition among pixels and limits precise pixel-wise control. To address these issues, we reformulate quantum image generation as coordinate-conditioned implicit function learning. Our method takes spatial coordinates and latent variables as inputs, uses a classical embedding network to generate input-dependent circuit parameters, and evaluates a variational quantum circuit at each coordinate. Pixel intensities are directly obtained from the expectation value of a dedicated color qubit, and a complete image is generated by querying all spatial coordinates. This design decouples image resolution from address-qubit requirements and avoids shared probability-normalization constraints across pixels. We further design a specialized variational quantum circuit to provide structural inductive bias for coordinate-conditioned generation. Simulated experiments on two benchmark datasets show that our method outperforms FRQI-based generation and PQWGAN in visual and quantitative quality while using fewer qubits, and also achieves better generation quality than the corresponding classical baseline.

Quaternion-valued neural networks and variational quantum circuits (VQCs) both derive local transformations from $\mathrm{SU}(2)$ geometry, yet their performance on classical supervised learning remains poorly understood. We compare real-valued, quaternion-valued, and quantum classification heads on identical frozen features across MNIST, FashionMNIST, and CIFAR-10. CIFAR-10 uses a learned 16-dimensional bottleneck and frozen ImageNet-pretrained ResNet18 features to separate architecture from representation quality. Quaternion classifiers match or approach real-valued baselines while outperforming shallow VQCs. On MNIST and FashionMNIST, quaternion networks nearly equal real-valued MLPs, whereas product-state VQCs show lower accuracy and higher cost. On CIFAR-10, quaternion networks retain 94--97% of real-valued performance and remain stable under a 32-fold increase in dimensionality. Product-state circuits underperform quaternion classifiers, while entanglement gives modest grayscale gains but reverses under pretrained CNN features (9.25 pp degradation vs.\ product-state). Fubini--Study/QFI natural gradients improve geometric alignment but not short-horizon loss reduction vs.\ Adam. A Friedman test on five-seed MNIST detects model differences ($χ^2=12.796$, $p=0.0051$, $n=5$), with Wilcoxon tests yielding large effect sizes ($d>5$) for QuatNet vs.\ quantum comparisons. For FashionMNIST and CIFAR-10, large effects ($d>2.0$) are the primary statistic given $n=3$. These results indicate that quaternion networks provide efficient, stable $\mathrm{SU}(2)$ alternatives to shallow VQCs on tasks lacking intrinsic quantum structure. Shared local $\mathrm{SU}(2)$ geometry and shallow entanglement are insufficient, within the regime studied, to confer practical quantum advantage. Conclusions are limited to shallow, measurement-limited circuits on such tasks.

The dynamical Lie algebraic (DLA) theory of variational quantum algorithms (VQAs) predicts commonplace exponentially vanishing loss and gradient variances for sufficiently deep parametrized circuits. In this work, we show that these predictions fail dramatically in the shallow-circuit (and particularly constant-depth) regime for the Quantum Approximate Optimization Algorithm (QAOA) applied to the maximum independent set (MIS) problem. In a large-scale numerical study across $\sim$23,000 problem instances, we find that barren plateaus are rare, while landscapes whose variances polynomially increase with system size---which we term "cragged terrains"---are common across graph families. This aggregate polynomial growth persists both for generic, low-symmetry random graphs and for highly symmetric vertex-transitive graphs, indicating that DLA-based variance predictions do not describe landscape scaling in this regime. As a stopgap alternative to the theory, we train empirical hardness models to predict instance-wise hardness metrics for QAOA-MIS. While these models generalize poorly, they nonetheless recover the correct landscape scaling class (barren plateau vs. cragged terrain) with high fidelity. Taken together, our results identify shallow QAOA for MIS as a prototypical setting in which asymptotic, unitary-design-centric predictions may be fundamentally insufficient to describe shallow variational quantum algorithms more broadly, emphasizing the need for more empirically-informed models of VQA loss landscapes.

Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.

Reinforcement-learning-based quantum architecture search (RL-QAS) repeatedly optimizes a variational quantum eigensolver (VQE) after extending a circuit, although circuit construction and action legality are deterministic and known. We introduce DreamQAS, a model-based RL framework that preserves these exact circuit dynamics and learns only the expensive post-VQE feedback. A recurrent randomized-prior ensemble predicts an oracle-free score relative to an empirical energy frontier and supports multi-step imagined policy learning over explicit legal circuits. Ranking-based activation, uncertainty-aware pessimism and truncation, and selective real-VQE verification form a reliability-controlled learning loop. Under a common 15,000-episode budget and frozen evaluation for the RL methods, DreamQAS has the lowest mean frozen-policy energy error on four of five molecular tasks and the second-lowest on one. At fine-error targets reached by all seeds of both methods, it uses 1.6x to 2.0x fewer real VQE calls on four tasks and 10.6x fewer on BeH2-8q. Counterfactual action-ranking utility increases across all five tasks, with a mean increase of 0.346 and a 95 percent confidence interval of [0.185, 0.507], while direct greedy and beam use of the same model does not recover the gains of imagined policy learning. Ensemble disagreement also improves risk-coverage over random rejection on all three probed tasks. These results establish a world-model design for QAS whose value lies in decision-useful feedback rather than exact energy prediction.

Variational quantum circuits have been central to many proposed near-term applications of quantum computing, but a growing body of evidence suggests that trainability and quantum advantage are fundamentally at odds: ansätze expressive enough to resist efficient classical simulation tend to exhibit barren plateaus, while structures that provably rule out barren plateaus typically render them classically simulable. We propose a stacked linear combination of unitaries (S-LCU) as a variational ansatz which provides a tunable trade-off between barren plateaus and classical simulability. Using a diagrammatic analysis, we bound the loss-landscape variance of the Free Fermion S-LCU, whose elements are fermionic Gaussian unitaries. We prove a variance lower bound of $Ω(1/(n k^{3l}))$, with a simulation cost of $O(k^{2l} n^3)$ using the best known classical algorithm, compared to a quantum gate complexity of only $O(lkn^2)$. The number of layers $l$ serves as a single dial that trades computational complexity against the rate of cost concentration. This offers practitioners a systematic method for constructing ansätze with a complexity-trainability trade-off that best suits their application and hardware.

Multivariate time series forecasting (MTSF) predicts future values of multiple variables from historical data. While quantum neural networks have been increasingly applied to this task, they typically rely on fixed local measurements, which restrict their expressivity. We propose MTSF-ANO, a simple hybrid model for MTSF that integrates variational quantum circuits with adaptive non-local observables (ANO). On the four ETT datasets, MTSF-ANO ranks first or second in MSE in 17 of 20 settings, improving over the strongest baseline by up to 20% on ETTh1, and outperforms or matches its fixed local observable counterpart across all settings. Our ablations show how the quantum circuit design and ANO non-locality affect performance. These results suggest that ANO is a promising direction for quantum time series forecasting.

Quantum Phase Estimation (QPE) is a foundational algorithm for molecular ground-state energy estimation, but its deep circuit requirements make direct hardware execution impractical on Noisy Intermediate-Scale Quantum (NISQ) devices. We present an analytically grounded variational surrogate framework in which a shallow Variational Quantum Circuit (VQC) is trained to reproduce the QPE measurement distribution without any quantum circuit simulation. The training target is computed entirely classically via the Dirichlet kernel, evaluated directly from the Full Configuration Interaction (FCI) ground-state energy, the ancilla qubit count, and the time evolution parameter, eliminating the exponentially scaling simulation bottleneck of prior surrogate approaches. We apply this framework to the hydrogen molecule (H$_2$) with a symmetry-tapered Hamiltonian, conducting a four-stage experimental investigation on IBM Quantum hardware. Stage 1 compares linear and full entangler topologies for the $R_Y$-$R_Z$-$CZ$ ansatz, with and without XpXm Dynamical Decoupling (DD), across four distributional metrics (Hellinger distance, fidelity error, total variation distance, Jensen-Shannon divergence), identifying the linear entangler as optimal. Stage 2 varies VQC layers ($p=1$ to $5$) for the linear-entangler ansatz, identifying single-layer depth as optimal under hardware noise. Stage 3 applies this configuration to the reduced $R_Y$-$CZ$ ansatz, comparing ideal and noisy simulator-trained parameters. A supplementary noise analysis at $p \in \{8,64\}$ characterizes the depth-dependent interplay between circuit depth and DD effectiveness. The framework enables faithful QPE mimicry using a linearly scaling VQC, recovering the ground-state energy within the chemical accuracy threshold (1 kcal/mol), constituting a scalable, hardware-efficient paradigm for QPE-based molecular energy estimation on NISQ devices.

Circuit cutting promises to scale quantum computations beyond current hardware, but variational quantum advantage also requires low cutting overhead, classical hardness, and trainability. We show that these properties are strongly constrained by entanglement geometry. Matrix product state (MPS) and tree tensor network (TTN) circuits with constant seam bond dimension can be cut with $O(1/\varepsilon^2)$ sampling overhead, but remain efficiently classically simulable, ruling out asymptotic quantum advantage within these families. By independently controlling seam and intra-block entanglement, we construct a two-block circuit family that remains cheaply cuttable while requiring a super-polynomial global MPS bond dimension, as supported numerically up to $n=100$. However, MPS hardness and trainability require incompatible depth regimes, $d=ω(\log n)$ and $d=O(\log n)$, respectively. Using magic rather than entanglement as the hardness resource avoids this conflict: shallow Clifford+$T$ circuits remain cuttable and trainable while their stabiliser-simulation cost grows exponentially with the $T$-count.

Variational Quantum Algorithms (VQAs) are a leading paradigm for near-term quantum computing, yet their training suffers from sensitivity to circuit depth, initialization, and landscape pathologies such as barren plateaus. We study \emph{progressive depth training} (PDT) -- a layerwise curriculum that trains a shallow circuit before appending new layers -- and identify a fundamental obstacle: fixed entangling gates (CNOTs) in hardware-efficient ansätze cause \emph{initialization shock}, an energy spike when new layers are added. We propose \emph{identity-paired progressive depth training} (IP-PDT), which appends forward/inverse block pairs -- each consisting of a standard rotation$+$CNOT block followed by its reverse -- that compose to the identity at initialization. Because the adjacent CNOT rings cancel, the effective circuit retains only \textit{a single entangling layer} surrounded by \textit{overparameterized local rotations}. We prove a simple \textit{Reachable Set Saturation Theorem}: under this construction the variational manifold expands exactly once (when post-entangler rotations are first introduced) and then \emph{saturates}; all subsequent depth increases provide pure overparameterization of single-qubit unitaries. Despite this saturation, progressive addition of rotation parameters can continue to improve optimization outcomes -- a phenomenon we term \emph{trainability beyond expressibility}. We formalize IP-PDT as a continuation method on nested manifolds, prove monotone energy guarantees under an acceptance rule, and connect energy error to ground-state fidelity through spectral-gap inequalities. A detailed resource analysis shows that IP-PDT achieves lower total gate cost than both baselines by eliminating most CNOT gates.

This paper presents a unified quantum-classical hybrid framework for multi-horizon time-series forecasting, introducing two model variants Quantum Reservoir Forecaster (QRC-F) and Variational Quantum Forecaster (VQF-F). The proposed framework investigates the complexity-fidelity trade-off of quantum forecasting under near-term NISQ hardware constraints. Continuous time-series signals are transformed into binary representations through uniform quantization and encoded into quantum states using angle encoding with parameterized RY rotation gates. Cross-channel entanglement layers capture dependencies among multiple variables. QRC-F utilizes a fixed random unitary quantum reservoir for stable, gradient-free temporal feature extraction, whereas VQF-F employs a trainable variational quantum circuit optimized through the parameter-shift rule to learn temporal and inter-variable patterns from Pauli expectation values. Both models replace computationally expensive quadratic self-attention with efficient linear transformations, reducing parameter complexity. A shared MIMO-based multi-horizon prediction head simultaneously generates forecasts across multiple horizons, avoiding error accumulation in recursive forecasting. Experimental evaluations on benchmark datasets, including ETTh1, ETTh2, ETTm1, ETTm2, Weather, electricity, and exchange-rate, demonstrate that VQF-F achieves superior training stability and parameter efficiency, while QRC-F provides enhanced robustness and circuit fidelity under quantum noise. The results establish a practical quantum-native forecasting framework with strong potential for deployment on near-term NISQ devices.

Hybrid quantum-classical machine learning workflows repeatedly evaluate many small parametrized circuits during training and model exploration. In this regime, framework dispatch and orchestration overhead often dominate runtime. Prior simulators accelerate execution but leave open the question of when compile-once specialization is the right choice for static variational circuits. We answer this question with VQCSim, a compile-once, PyTorch-native statevector execution path with native autograd. In a systematic MQT Bench study, VQCSim compiles all tested static circuits and provides 87.7% end-to-end semantic validation. Across a five-GPU evaluation set, VQCSim delivers pooled median speedups of 4.49x for native inference and 26.78x for native training, while retaining a 3.31x advantage under matched finite-difference training. Ablation identifies native autograd as the dominant source of acceleration (27.6x), with compile-once caching and batch vectorization contributing additional gains. The speedup trades higher GPU memory (VQCSim is memory-limited at the high end) for lower runtime. We derive a hardware-aware regime map and release vqcsim-oracle, an open-source backend selector with 91.1%-97.7% top-1 agreement (including cross-GPU transfers), enabling automatic simulator selection in QML design loops.

We propose a unified meta-decoding framework for quantum error correction that learns syndrome-to-recovery mappings across multiple stabilizer codes and noise settings, without requiring separate decoders for each configuration. The benchmark includes FiveQubit, Steane, Planar3x3, and Planar5x5 codes, four noise families, and five evaluation regimes: interpolation, unseen-p transfer, unseen-noise transfer, few-shot unseen-code adaptation, and few-shot held-out-size adaptation. We compare a classical Meta-MLP teacher-trained baseline with variational quantum circuit (VQC) meta-decoders selected through hardware-aware quantum architecture search over qubit count, circuit depth, and entangling topology. The Meta-MLP achieves teacher-label accuracies of 0.9993, 0.9118, 0.9342, 0.6304, and 0.7548 across the five regimes, while the hardware-aware VQC achieves 0.9400, 0.8495, 0.8415, 0.5678, and 0.7143. However, logical-level evaluation shows that high teacher-label accuracy alone is insufficient in the most challenging Planar5x5 setting. During interpolation, the raw logical-failure ratios relative to the teacher are 12.08 and 25.91 for the Meta-MLP and VQC, respectively, whereas confidence-gated fallback reduces them to 1.71 and 1.11. These results support confidence-aware selective recovery rather than unconditional teacher replacement.

We study the integration of variational quantum circuits (VQCs) into diffusion models through a squeeze-and-excitation (SE) channel-modulation scaffold that isolates the quantum contribution. Using a role-matched classical control and multi-seed significance testing across DDPM and latent diffusion on MNIST and CIFAR-10, with a score-based NCSN study on MNIST, we find that quantum cores achieve comparable mean FID to the classical control across DDPM and latent diffusion, while paired sampling-seed tests for EfficientSU2 detect no statistically significant difference. Although the quantum cores use $4.5$--$9\times$ fewer core parameters than the role-matched control, parameter-matched classical controls attain comparable mean FID, so the experiments do not establish a quantum parameter-efficiency advantage. We further identify a structural failure in score-based NCSN: the unbounded score target, proportional to $1/σ$, drives angle-embedding inputs far beyond the $2π$ period of rotation gates, causing phase aliasing and collapse of the quantum modulator. A bounding transformation, $θ\leftarrow π\tanh(\cdot)$, maps inputs to the non-aliasing domain and substantially improves both quantum cores. Since all circuits are classically simulated at a few-qubit scale, we do not claim quantum advantage. Instead, the study provides a fair-comparison protocol for quantum-enhanced generative models and a mechanistic account of when and why angle embeddings fail.

Federated learning (FL) enables collaborative model training across distributed devices without sharing raw data, making it suitable for privacy-sensitive robotic sensing applications. However, multi-agent systems generate heterogeneous and non-independent and identically distributed (non-IID) multimodal sensor streams that degrade conventional FL algorithms, while classical fusion modules introduce substantial parameter overhead and communication cost. This paper proposes QFedAgent, a hybrid quantum-classical personalized FL framework for multi-agent activity recognition. The approach integrates a variational quantum circuit fusion module that models accelerometer--gyroscope interactions through quantum state encoding and entanglement, requiring only 72 quantum rotation parameters versus 33K in classical multi-layer perceptron-based fusion, achieving approximately 10x total parameter reduction. Experiments on the OPPORTUNITY dataset under subject-based non-IID partitions demonstrate 97.7% mean test accuracy, confirming that parameter-efficient quantum fusion remains competitive with conventional federated baselines.

The geometric and topological structure of quantum cost landscapes (QCLs) governs the optimization and thus the predictive power of variational quantum algorithms (VQAs). We systematically analyze ravines - low-cost paths connecting local minima - using an adapted version of the nudged elastic band (NEB) algorithm, a method originating from theoretical chemistry. By training quantum neural networks (QNNs) to classify the concentratable entanglement of quantum states, we apply the NEB algorithm and numerically identify ravine structures in QCLs of hardware-efficient ansatzes. Beyond visualizing these ravines, we construct an ensemble prediction framework by averaging predictions from QNNs parameterized along the low-cost NEB path. We introduce a resource-light pre-training metric which quantifies local-prediction variability and serves as a strong performance indicator for VQAs, even beyond the scope of this study. When base classifiers are drawn from circuit and weight initializations exhibiting high local-prediction variability, the quantum-based NEB ensembles outperform both classical and naive quantum alternatives. Moreover, a complexity analysis shows that leveraging the ravine-like structure of QCLs with the QNN NEB approach substantially reduces computational costs compared to naive QNN ensembling. A depth and qubit scaling analysis indicates that ravines persist across both scalings, and that, despite the expected growth in resource requirements with the qubit scaling, the NEB approach also accelerates convergence over the naive alternative.

We investigate Gaussian process (GP) bandit optimization with quantum kernels, assuming the mean reward function lies in the reproducing kernel Hilbert space (RKHS) induced by the quantum kernel. This setting is motivated by NISQ-era tasks such as quantum control, state preparation and variational quantum algorithms. While quantum kernels can offer a `quantum advantage' via domain-specific inductive biases, naïvely using full, high-dimensional kernels increases model complexity and information gain, leading to higher cumulative regret and poor learnability. To address this, we propose projected quantum kernels and classical kernel approximation techniques that reduce feature dimensionality while preserving key quantum properties. Using these approximate kernels, we develop misspecified GP bandit algorithms and derive regret bounds that characterize the trade-off between approximation error and information gain. The regret bounds provide principled guidance for selecting the optimal model complexity. Empirically, our methods outperform full quantum kernels in sample efficiency, while substantially reducing computational overhead, enabling scalable GP optimization for quantum-native applications.

Visual environments are a demanding setting for quantum reinforcement learning (QRL): high-dimensional observations, unstable RL optimisation, and constrained variational quantum circuits (VQCs) are difficult to train jointly. This paper studies knowledge distillation (KD) as a staged hybridisation strategy for visual QRL. Instead of training a hybrid visual agent end-to-end from pixels, we first train a classical visual teacher, freeze its encoder as a feature interface, and distil the teacher's policy behaviour into compact downstream heads. These heads can be classical or VQC-based, enabling small quantum-compatible students to be evaluated under the same frozen representation as compact classical controls. We evaluate the pipeline on CartPole Pixels and Acrobot Pixels. The results show that staged KD enables shallow VQC heads to acquire non-trivial visual-control behaviour in settings where direct pixel-based training would be substantially more difficult. Angle-encoded VQC heads retain near-teacher performance, while amplitude-encoded heads push compactness to an extreme regime, at the cost of greater fragility, stronger budget sensitivity, and higher simulation time. Overall, staged KD reframes visual QRL as a compact-head learning problem, opening a practical route for training small quantum-compatible policies outside the standard end-to-end RL loop.

Graphs provide a natural language for relational data in chemistry, biology and optimisation. Graph neural networks (GNNs) have driven much of the recent progress in learning from such data through message passing, a single primitive that generalises convolution and attention. Quantum counterparts have been proposed, but with limited connection to message passing and few guarantees on performance or scalability. More broadly, the trainability of variational quantum circuits is a recognised bottleneck for their wide applicability, and pre-training has emerged as one way to address it. Yet for a quantum model to be useful, it must offer expressivity guarantees along with demonstrable scalability. Here we show how a quantum graph neural network can be built to perform message passing, to be permutation equivariant, and to sit at a chosen level of the Weisfeiler-Leman hierarchy, the standard measure of how finely a model can tell graphs apart. We show that, as for classical GNNs, the training can be done first on small graph instances, allowing for a pre-training that can mitigate usual training issues, and its output can be read out at a cost that stays low as the graph grows. We validate the framework in large-scale simulations of up to 56 qubits across three datasets, on synthetic graphs that ordinary message passing cannot separate, on molecular property prediction, and on the travelling salesperson problem. Our framework opens a path for near-term quantum algorithms with theoretical guarantees and practical scalability, bringing the principles of graph learning into quantum circuit design.

Variational Monte Carlo (VMC) is a central algorithm in electronic structure theory and has gained renewed importance through modern neural-network ansätze such as FermiNet. At its core, VMC seeks ground states by minimizing the Rayleigh quotient by stochastic optimization. In this work, we show that the resulting stochastic optimization problem is intrinsically governed by the nodal geometry of the underlying wave function. More precisely, we establish that properties of the nodal set determine the integrability of the local energy and gradient estimators that drive VMC. For broad and practically relevant ansatz classes, including Slater-Jastrow wave functions with variable-exponent Slater-type orbitals, we prove that these estimators are generically heavy-tailed and fail to admit higher moments. At the same time, for general analytic ansätze, we prove weak moment bounds for the relevant estimators and identify precise low-moment regimes, showing how generic and degenerate nodal structures lead to different integrability thresholds. Building on this analysis, we introduce a new robust variant of VMC $\unicode{x2013}$ coined PS-Clip-VMC $\unicode{x2013}$ which is based on clipping both the local energy and the gradient random variable. We prove that PS-Clip-VMC converges both in expectation and with high probability in the weak moment regime of VMC. Preliminary experiments for training FermiNet on Atoms with up to 18 electrons suggest that PS-Clip-VMC is significantly more robust than standard methods.

Recent advances in quantum computing and machine learning have motivated the development of quantum models for sequential data processing. In this paper, we propose a Recursive Quantum Long Short-Term Memory model, or Recursive QLSTM, which extends QLSTM through metacore-based recursive constructions. We numerically test the model under different input sequence lengths, metacore designs, and recursive rules, and identify the best-performing architecture among these variants. For this selected model, we further provide theoretical arguments explaining why its recursive structure improves temporal information propagation and enhances learning performance. Our results suggest that Recursive QLSTM offers a flexible and effective framework for quantum recurrent learning over input time series of various lengths.

We present a systematic study of von Neumann entropy estimation in multi-qutrit quantum systems using two complementary approaches: variational quantum algorithms (VQAs) and classical convolutional neural networks (CNNs), evaluated using an ideal (noise-free) quantum simulator. For systems up to three qutrits, we construct and evaluate 11 hardware-efficient SU(3)-inspired ansatzes. A parameter sweep shows that estimation accuracy is primarily determined by the number of trainable parameters, provided sufficient entanglement is present. Based on this study, we fix the parameter count to approximately 120 for subsequent experiments, observing that increasing entangling-gate counts beyond a threshold yields only marginal improvements. For larger systems (two to five qutrits), we use a CNN trained on measurement outcomes from tensor-product mutually unbiased bases. The model achieves accurate and stable predictions and exhibits a systematic improvement in performance with system size, with the highest errors for two-qutrit systems and the lowest for five-qutrit systems. Notably, using only 12.5% of the measurements required for full state tomography is sufficient to reach 90th-percentile absolute errors of approximately 0.13-0.16 nats for both four- and five-qutrit systems. The CNN model is also robust to shot noise and generalizes well to out-of-distribution states. Overall, within the simulated settings studied here, our results indicate a transition in practical methods: VQAs are effective for small systems, while CNN-based estimators offer improved scalability and robustness for larger qutrit systems.

The Quantum Approximate Optimization Algorithm (QAOA) is a leading variational algorithm for combinatorial optimization on near term quantum devices. As circuit depth increases, the number of optimization parameters grows, making the search landscape increasingly nonlinear and difficult to optimize. Previous studies have shown that optimal QAOA parameters often lie on a low dimensional manifold that can be approximated using Principal Component Analysis (PCA) at shallow circuit depths. However, the effectiveness of PCA decreases at higher depths because the underlying parameter manifold becomes increasingly nonlinear. In this work, we investigate Kernel Principal Component Analysis (KPCA) with a radial basis function kernel as a nonlinear dimensionality reduction technique for QAOA parameter optimization. The model is trained using 200 graphs from each of 3 graph families, namely Erdos-Renyi, Barabasi-Albert, and Watts-Strogatz, with graph sizes ranging from 7 to 10 nodes. Performance is evaluated on 30 test graphs containing 12 nodes at circuit depths 1, 2, 4, and 8. Experimental results demonstrate that KPCA consistently outperforms PCA at deeper circuit depths across all graph families. At depth 8, KPCA achieves approximation ratios above 0.86, while PCA declines to approximately 0.81 to 0.83. Both methods reduce the number of quantum circuit evaluations by more than 93 percent relative to unrestricted QAOA optimization. These findings suggest that nonlinear kernel methods more effectively capture the structure of the QAOA parameter manifold and provide a practical approach for scaling variational quantum optimization to deeper circuits.

We develop a framework for the information discarded by machine learning models whose inputs carry a Lie group action. Given a representation $π$ of a Lie group $G$ on a space $V$ and a learned function $f\colon V \to \mathbb{R}$, we define two objects measuring the symmetry invisible to $f$. The null fiber at a point $x \in V$ is the set $N_G(f,x) = \{g \in G : f(π(g^{-1}) \cdot x) = f(x)\}$ of group elements whose inverse action on $x$ is undetectable by $f$. When $N_G(f,x)$ is independent of $x$, it coincides with the stabilizer $\mathrm{Stab}_G(f)$, the largest subgroup of $G$ under which $f$ is invariant. For smooth maps to $\mathbb{R}$, the preimage theorem guarantees that null fibers have dimension at least $\dim G - 1$ at generic inputs, regardless of architecture. For compact groups acting on themselves, the Peter--Weyl theorem yields a spectral characterization of both objects in terms of the Fourier coefficient matrices of $f$. We show that null fiber elements can be computed efficiently via Newton iteration on the orbit map, at a cost comparable to a few gradient evaluations. Applications to data masking, model fingerprinting, and privacy-preserving computation are developed and tested experimentally on molecular property prediction under $\mathrm{SO}(3)$ and spherical image classification under the Möbius group $\mathrm{PSL}(2, \mathbb{C})$. The framework applies uniformly to classical neural networks and variational quantum circuits.

Recent advances in Machine Learning have transformed numerous industrial sectors, yet classical paradigms face fundamental limitations: rapidly growing data volumes, rising computational costs, significant energy consumption, and the physical scaling limits of conventional hardware architectures. Quantum computing has emerged as a promising computational paradigm to address these challenges, giving rise to the field of Quantum Machine Learning (QML). In this thesis, the theoretical foundations of QML are investigated, with a focus on near-term and future practical applications. Three central challenges are addressed: the trainability of variational quantum circuits, their expressivity, and their resistance to efficient classical simulation. The trainability of Hamming-weight preserving variational quantum circuits is first studied, and theoretical guarantees are established that resolve an open conjecture on the absence of barren plateaus for this circuit family. Subspace-preserving QML algorithms are then introduced, including photonic circuits and quantum convolutional neural networks, and are designed to mimic classical ML subroutines while offering polynomial quantum advantage. Finally, variational quantum circuits are analyzed as quantum Fourier models, and a framework is derived to jointly characterize expressivity and trainability, from which conditions are obtained under which quantum models provably separate from their classical counterparts. These contributions are intended to advance the theoretical roadmap for harnessing near-term and future quantum technologies in real-world applications.

The design of high performing quantum circuits remains largely dependent on human expertise. We introduce an autonomous agentic framework that employs large language models (LLMs) to conduct iterative quantum circuit designs under explicit design constraints. Our system integrates seven components: Exploration, Generation, Discussion, Validation, Storage, Evaluation, and Review. These components form a closed-loop workflow that combines web-based knowledge acquisition, literature-grounded critique, executable code generation, and experimental feedback. We evaluate the framework on two tasks: quantum feature map construction for quantum machine learning and ansatz generation for variational quantum eigensolver applications in quantum chemistry. In image classification benchmarks, the best generated feature map outperforms representative quantum feature maps and, when scaled to larger qubit counts, surpasses the classical radial basis function kernel. In molecular ground state estimation across seven molecules, the generated ansatz attains competitive accuracy with widely used chemically inspired and hardware-efficient constructions while satisfying the imposed scaling constraints. These results establish LLM driven agentic system as a viable paradigm for automated quantum circuit design and illustrate how AI systems can participate in iterative scientific optimization workflows across scientific domains.

Generative models are increasingly used to capture correlations in many-body systems, but the representations they learn remain largely opaque to physical interpretation. Here, we establish an intuitive criterion that quantifies the capacity of a variational autoencoder (VAE) to faithfully reconstruct the joint probability distribution of a many body system. In a nutshell, a bound on the VAE capacity is obtained by comparing the rate of the latent channel to the bipartite mutual information of the data. Using this bound, we show that the conditionally independent decoder of any successful VAE is structurally identical to a finite-size mean-field factorization. Hence, a successful reconstruction is direct evidence for a latent mean-field theory and the microscopic parameters of that theory can be read off the trained decoder. We validate these conclusions on a hierarchy of solvable models with scalar (Curie-Weiss), vector (Hopfield) and tensor (Maier-Saupe) order parameters, recovering the full Hopfield pattern matrix from equilibrium samples alone. We find that, when applied to Salamander retinal recordings, a two-latent VAE reproduces the population statistics with only two effective collective variables allowing us to recover the `stored patterns' of the neural population and write a generalized Hopfield model which correctly models the experimental data.

Efficient quantum error correction is essential for the advancement of quantum computing. We propose a quantum neural network with a global structure that reduces the number of unitary matrices required in quantum circuits. This approach resulted in a 97% reduction in training time and up to a 25% improvement in the training completion rate, ultimately achieving a 100% success rate in training while surpassing the error correction performance reported in previous studies. In addition, we demonstrated the enhanced robustness of quantum error correction against internal network noise. Moreover, the fidelity of quantum error correction under internal network noise increased by up to 15% due to the reduced computational load.

Quantum combinatorial optimization offers theoretical advantages for complex financial modeling, but physical implementation on Noisy Intermediate Scale Quantum (NISQ) devices is severely constrained by hardware topology. This study presents a hardware benchmarking analysis between a Hardware Efficient Variational Quantum Neural Network (HE-VQNN) and the Warm Start Quantum Approximate Optimization Algorithm (WS-QAOA) for a hybrid Mean Variance and Conditional Value at Risk (CVaR) portfolio objective. By implementing a novel classical quantum hybrid proxy matrix to bypass the CVaR auxiliary qubit bottleneck, we map up to 16 assets from the NIFTY 50 index onto an IBM heavy hex processor. We systematically quantify algorithmic resilience to the "SWAP tax" incurred during routing. Empirical results reveal a critical operational trade-off: WS-QAOA provides exact theoretical mapping but suffers catastrophic hardware decoherence due to exponential nonlocal gate overhead. Conversely, HE-VQNN preserves hardware coherence but lacks the mathematical expressibility to capture dense tail risk asset correlations. This study exposes the limitations of dense financial optimization on current architectures forces an nonviable choice between algorithmic inexpressibility and hardware decoherence. This is indicative of a deeper limitation as to what can and cannot be done with NISQ computers lacking in all-to-all connectivity.

Large machine learning models benefit substantially from multimodal inputs that provide a complementary view of the same example. We introduce QUIVER (QUantum-Informed Views for Enhanced Representations, a paradigm that enriches classical data-driven features with a quantum Fisher view: a geometrically motivated, basis-independent summary of higher-order correlations captured by a variational quantum circuit (VQC) trained to perform the same task. Unlike classical feature augmentation, the quantum Fisher information matrix encodes the intrinsic geometry of the learned quantum state manifold. While this feature map, motivated by quantum information theory, is ordinarily non-trivial to model classically, it can surface statistical structure that additional classical data or model capacity finds difficult to learn. This makes the quantum Fisher view a genuinely complementary modality rather than a redundant one. We demonstrate that QUIVER improves standard performance metrics on two benchmark datasets from very different fields: QM9 for predicting molecule properties, and JetClass for predicting jet flavor at the Large Hadron Collider (LHC). The core contribution, however, is domain-agnostic: the quantum Fisher view can be fused into a broad class of model architectures via targeted modifications to the base architecture, to incorporate information about the quantum geometry of the problem. These results demonstrate that quantum-geometric features, extracted from simulated variational circuits, can deliver measurable value for standard machine learning tasks, well before the advent of fault-tolerant quantum hardware.

Training Variational Quantum Circuits (VQCs) under Noisy Intermediate-Scale Quantum (NISQ) constraints introduces severe computational limitations: classical statevector simulation memory scales exponentially ($\mathcal{O}(2^n)$), and global cost functions suffer from barren plateaus where gradient variance decays exponentially ($\mathcal{O}(1/2^n)$). This paper introduces and evaluates the Quantum Algorithm for Distributed Reduction of Entanglements (QADR), a hybrid quantum-classical machine learning framework that decomposes a global $n$-qubit VQC into localized sub-circuits operating approximately within the causal light cones of individual target qubits. QADR reduces classical simulation memory scaling from $\mathcal{O}(2^n)$ to $\mathcal{O}(n \cdot 2^{2d+1})$ for a light cone radius $d$, while naturally mitigating global barren plateaus. We benchmark QADR against standard global VQCs, Support Vector Machines (SVM), and two customized classical parameter-matched neural networks (CANN and PMNN) on the MNIST dataset and the high-dimensional NASA IMS wind turbine drivetrain diagnostic task. QADR demonstrates excellent scalability, operating successfully at $n_{\text{features}}=2000$ where standard global VQCs crash due to memory exhaustion, while matching or exceeding the performance of optimized classical architectures.

We train a pair of autoregressive models to construct zero-mean control variates to mitigate the sign problem in quantum Monte Carlo simulations. The two autoregressive networks are confined to the positive- and negative-sign sectors with strictly disjoint support, and each is exactly normalized over its sector. Their difference is therefore structurally zero-mean, providing an unbiased auxiliary observable whose correlation with the sign estimator controls the variance reduction. We implement the method within the stochastic series expansion framework, which we extend to frustrated lattices by developing an incremental loop-topology update. Sign-ergodic sampling is achieved through a twist channel, which is the unique sign-changing mechanism on non-bipartite lattices. We implement the control variates as autoregressive transformers with an end-of-sequence parity mask that enforces exact sign-sector resolution, while the incremental loop-count change and cumulative frustration parity are incorporated as topological features. On the triangular-lattice Heisenberg antiferromagnet, we benchmark the method in the small-$N$ limit. The control variate reduces the standard error of the average sign by up to an order of magnitude and that of the energy estimator by a factor of three to five, remaining effective even when the average sign drops below $10^{-3}$. This work lays out the framework and provides a proof-of-principle demonstration that autoregressive control variates can effectively mitigate the sign problem. Scaling to larger systems with physics-informed architectures is the subject of future work.

Statistical downscaling is a crucial component of the weather modeling field, where high-resolution outputs must be reconstructed from coarse-resolution inputs with the full cost of dynamical refinement. In this work, we investigate a hybrid quantum-classical corrective diffusion model for probabilistic statistical downscaling of weather fields. The proposed model inserts variational quantum circuit layers into the most compressed bottleneck of the diffusion UNet while leaving the regression branch fully classical. This placement tests whether quantum circuits can act as compact nonlinear feature maps for latent-channel mixing. We evaluate intra-channel and cross-channel ansätze on 10m wind components. On the 2020 validation set, the hybrid models remain stable, preserve the large-scale spatial organization of the generated wind fields, and improve both MAE and CRPS relative to a classical corrective diffusion model in several configurations. Structural diagnostics further show that the hybrid variants preserve kinetic-energy spectra and windspeed distributions similar to its classical counterpart while producing controlled changes in tail behavior, extreme-windspeed localization, and joint wind field components structure. Backend studies on the 2020 validation set show negligible impact from simulated device noise at the tested circuit scale, whereas real-hardware deployment remains limited by qubit availability and execution fidelity. The 2021 out-of-distribution test shows that these in-distribution gains do not transfer uniformly under temporal shift, revealing a generalization gap that motivates future mitigation through stabilization and regularization. These results show that bottleneck-level quantum hybridization can make a nontrivial contribution to weather statistical downscaling, while also highlighting that circuit scale and hardware deployment remain key limiting factors.

Variational Quantum Algorithms (VQAs) potentially offer a pathway to practical quantum advantage, but their optimization is heavily hindered by barren plateaus and numerous local minima. While classically simulable Clifford circuits can warm-start VQAs to accelerate convergence, existing heuristic-based initialization methods struggle to scale within vast combinatorial search spaces. To overcome this bottleneck, we propose CRiSP (a Clifford Reinforcement Learning agent for State Preparation), a framework that formulates discrete prefix selection as a sequential decision-making problem. CRiSP utilizes Neural-Guided Monte Carlo Tree Search, driven by a Transformer-based policy trained via self-play, to insert learned Clifford gates before fixed parameterized rotations. This enables the construction of high-quality initial states entirely through polynomial-time classical stabilizer simulation without altering the underlying circuit architecture. By integrating a curriculum learning strategy that progressively expands the search horizon, the agent efficiently scales to deep circuits. Evaluated on QAOA benchmarks of up to $22$ qubits and $1{,}370$ parameters, CRiSP outperforms state-of-the-art Clifford initialization methods by a mean of $3.17\times$ (max $45.02\times$) in average energy accuracy and $2.44\times$ (max $16.01\times$) in best-achieved energy accuracy. Assessments on VQE tasks further demonstrate the framework's robustness and generalizability.

Quantum Federated Learning (QFL) inherits the core vulnerability of federated optimization to malicious clients, while also introducing an attack surface from variational circuit training and measurement-driven gradients. This work proposes a novel CircUit-Level backdoor Threat (CULT) model that formalizes four stealthy attacks by exploiting quantum-aware mechanisms, including Grover, Pauli, Bit-flip, and Sign-flip. By enabling malicious clients on both in-training and post-training surfaces, these attacks can critically undermine the learning process. We establish a rigorous theoretical foundation to demonstrate attack stealthiness under standard smoothness assumptions. Experiments on the MNIST and CIFAR-10 datasets with non-IID splits and varying fractions of malicious clients show that even a single malicious client can induce severe accuracy degradation under FedAvg aggregation. While popular defenses, including Krum, Multi-Krum, FoolsGold, FLGuardian, and Mud-HoG, reduce degradation in many regimes, they fail to eliminate worst-case failure cases, where accuracy drops up to 50%. The experimental analysis further reveals that under the CULT model, malicious updates effectively mask their presence by staying close to benign norms, thereby helping attackers evade detection.

We propose a variational quantum classifier operating on high dimensional deep representations via amplitude encoding, stabilized by a learnable classical pre encoding layer.By combining normalized amplitude embeddings with bounded quantum observables, the resulting model induces a structured and smooth hypothesis class with controlled sensitivity to input variations. Model reliability is assessed using SAFE-AI metrics derived from the Cramer von Mises divergence, enabling consistent evaluation across accuracy, robustness, and explainability dimensions. Empirical results show that the proposed quantum model provides competitive predictive performance compared with strong classical baselines while exhibiting a more balanced SAFE reliability profile, with improved robustness to noise and stability under structured feature removal. These findings suggest that variational quantum circuits offer a principled mechanism for stability oriented SAFE learning in safety critical settings.

Adaptive Non-local Observables (ANOs) have shown that making quantum observables dynamic can substantially enlarge the function space of Variational Quantum Algorithms, partly shifting hardware demands from circuit synthesis to measurement design. However, this advantage is accompanied by a steep increase in the number of parameters, as well as the classical optimization cost for varying general Hermitian observables. We propose a special form of ANO that significantly reduces this burden by considering only diagonal observables paired with quantum circuits. Mathematically, this is equivalent to the full ANO of a large parameter space since diagonal matrices are canonical representatives of the ANO space modulo unitary similarity. As a result, Diagonal ANO retains the same capability of full ANO while reducing $k$-local observable complexity from $O(4^k)$ to $O(2^k)$ and lowering the corresponding measurement-side classical computation. In this sense, diagonal ANO preserves much of the benefit of full ANO while encompassing conventional VQCs as a special case.

We present an empirical evaluation of quantum entanglement in agent coordination within quantum multi agent reinforcement learning (QMARL). While QMARL has attracted growing interest recently, most prior work evaluates quantum policies without provable baselines, making it impossible to rigorously distinguish quantum advantage from algorithmic coincidence. We address this directly by evaluating a decentralized QMARL framework with variational quantum circuit (VQC) actors with shared entangled states. In the CHSH game, which has a mathematically proven classical performance ceiling of 0.75 win rate, we show that entangled QMARL agents approach the Tsirelson limit of 0.854, providing clear evidence of their quantum advantage. We show that unentangled quantum circuits match the classical baseline, confirming that entanglement and not the quantum circuit itself is the active coordination mechanism. We also explore the effect of specific entanglement structures, as some Bell states enable coordination gains while others actively harm performance. On cooperative navigation (CoopNav), QMARL without entanglement achieves $\sim2\times$ improvement in success rate over classical MAA2C ($\sim$0.85 versus $\sim$0.40), with the hybrid configuration, quantum actor paired with a classical centralised critic, outperforming both fully classical and fully quantum solutions. We present our experimental analysis and discuss future work.

Variational Quantum Algorithms (VQAs) are a leading approach to exploiting near-term quantum hardware, leveraging parameterized quantum circuits and classical optimization to achieve advantage. Despite their promise, the practical deployment of VQAs is challenged by the difficulty of designing quantum circuit architectures that balance expressivity, trainability, and hardware constraints. Existing evolutionary-based quantum neural architecture search methods address these challenges but suffer from high computational costs due to repeated training of candidate circuits. In this work, we identify a setting in which the Gram matrix of the Quantum Neural Tangent Kernel converges. Building on this observation, we design a zero-shot surrogate model to estimate candidate performance without full training, significantly accelerating the architecture search process. Using this surrogate, we propose MZeQAS, a Monte Carlo Tree Search (MCTS)-based Zero-Shot Quantum Neural Architecture Search framework for VQAs. By integrating proxy-based performance estimation with MCTS exploration, MZeQAS efficiently discovers high-performing architectures. Experimental results demonstrate that MZeQAS outperforms existing approaches in terms of both search efficiency and solution quality, providing a scalable and effective framework for advancing VQA deployment on noisy intermediate-scale quantum devices.

Partial differential equations (PDEs) are central to modeling physical and engineering systems, but repeatedly solving parametric PDEs remains computationally expensive. Operator learning enables fast surrogate inference, yet typically requires large input-output paired datasets generated by costly high-fidelity PDE solvers. Unsupervised operator learning frameworks alleviate data dependency but remain hindered by computational bottlenecks. To address this, we propose Neural Variational Quantum Linear Solver (NVQLS), the first hybrid quantum-classical operator learning framework leveraging the Legendre--Galerkin weak formulation. We critically resolve the sign ambiguity in VQLS energy minimization, preventing erroneous solution representations. Additionally, we introduce a neural embedding, a novel encoding scheme to map varying forcings and PDE coefficients into parameterized quantum circuit representations. These structural innovations provide theoretical computational complexity advantages under efficient state preparation schemes, while achieving superior accuracy compared to a representative classical baseline. Validations on 1D and 2D parametric PDEs under diverse boundary conditions demonstrate NVQLS's capability to simultaneously process varying inputs, offering a scalable unsupervised approach to quantum-enhanced operator learning.

Classical generative adversarial networks (GANs) have been applied to generate adversarial network traffic capable of attacking intrusion detection systems, but they suffer from shortcomings such as the need for large amounts of high-dimensional datasets, mode collapse, and high computational overhead. In this work, we propose a hybrid quantum-classical GAN (QC-GAN) framework where a variational quantum generator is used to generate synthetic network traffic flows mimicking malicious traffic using latent representations. Instead of sampling classical noise vectors, we encode the latent vector (the hidden features) as a quantum state, which is the basis for claiming more expressive latent representations and reducing computational overhead. A classical discriminator will be trained on real-world datasets (UNSW-NB15) and the proposed QC-GAN-generated fake network flows. In this configuration, the generator aims to minimize the discriminator's ability to distinguish real from fake traffic, while the discriminator aims to maximize its classification accuracy, in an iterative manner. In our attack model, we assume that the attacker is a state actor with access to limited quantum computing power, whereas the discriminator is chosen to be classical, as will likely be the case for most end users and organizations. We test the generated flows using classical intrusion detection system (IDS) models, such as a random forest classifier and a convolutional neural network-based classifier, for their ability to bypass the detection process. This work aims to highlight the possibilities of quantum machine learning as a means of generating advanced attack flows and stress testing classical IDS. Lastly, we further evaluate how hardware-based noise affects these attacks to offer a new perspective on IDS, highlighting the need for a quantum resilient defense system.

Variational quantum circuits with angle encoding implement truncated Fourier series, and architectures arranging $N$ qubits with $L$ encoding layers each -- sharing encoding budget $E = NL$ -- generate identical frequency spectra, identical frequency redundancy, and require the same minimum parameter count for coefficient control. Despite this equivalence, trainability varies substantially with architecture shape $(N,L)$ at fixed $E$. We identify structural rank deficiency of the coefficient matching Jacobian $J$ as the mechanism responsible. For serial single-qubit architectures, we prove $\mathrm{rank}(J) \leq 2L+1$ regardless of parameter count $P$, with $\dim(\ker J) \geq P-(2L+1)$ growing without bound -- a phenomenon we term \emph{structural gradient starvation}: a growing fraction of parameters become structurally decoupled from the loss as $P$ increases at fixed $L$. Parallel architectures avoid this via independent phase trajectories, ensuring $σ_{\min}(J^{(\mathrm{par})}) > 0$ generically for $P \leq 2E+1$, so no parameter lies in $\ker J$. For practitioners, we further show that the two natural routes to increasing parameter count have fundamentally different effects: adding feature map (FM) layers monotonically strengthens the Jacobian QFIM eigenvalue spectrum and achieves $R^2 \geq 0.95$ with $1.6$--$2.2\times$ fewer parameters than adding trainable blocks across all tested architectures, while trainable blocks improve training only through the classical interpolation mechanism with no quantum-specific benefit.

Distributed quantum algorithms offer a promising pathway to scale variational quantum algorithms beyond the constraints of noisy intermediate-scale quantum hardware. However, existing approaches implicitly assume a trusted entanglement-sharing layer across quantum processors. We show that this assumption introduces a fundamental vulnerability: adversarial perturbations of shared entanglement induce structured gate-level noise that directly impacts quantum learning. We develop a framework that maps entanglement-level perturbations to gate-level noise via an explicit Kraus representation. To quantify their impact, we introduce Kraus expressibility, a metric that generalizes unitary expressibility to noisy quantum channels. We then establish a trade-off between Kraus expressibility and trainability of noisy quantum circuits through gradient variance analysis. Our analysis reveals that an adversary can manipulate Kraus expressibility to maintain sufficiently large cost gradients (avoiding barren plateaus) while systematically biasing optimization toward incorrect solutions. We validate these findings through numerical simulations, demonstrating adversarial degradation of expressibility and trainability.

While parameterized quantum computations have shown success in standard reinforcement learning (RL), whether these advantages adapt to hierarchical RL (HRL) remains a critical open question. This work demonstrates that variational quantum circuits (VQCs) can effectively enhance HRL agents based on the option-critic architecture. Evaluated in standard environments, a hybrid HRL agent with a quantum feature extractor outperforms classical baselines while using fewer parameters. We also identify an architectural bottleneck: using VQCs for option-value estimation severely degrades learning. Further ablations reveal how quantum circuit design affects performance. Our work establishes design principles for parameter-efficient hybrid HRL agents.

This paper presents Quantum Gatekeeper, a context-bound image steganography framework where successful payload recovery depends on both cryptographic decryption and the reconstruction of a precise extraction path. The system integrates lossless least significant bit (LSB) embedding with a deterministic variational quantum circuit (VQC)-derived gate key, multi-factor contextual binding, and authenticated encryption. Payload extraction is contingent upon four requisite factors: a password, a shared secret, a user-supplied context string, and a reference image signature. Any deviation in these factors causes the system to read from an incorrect pixel sequence or fail authentication, resulting in silent rejection rather than partial disclosure. The proposed method derives a gatecontrolled extraction key from a seed-conditioned variational circuit, with parameters generated via cryptographic hash expansion and context-dependent image features. To ensure encode/decode consistency, the cryptographic key path is generated via exact statevector simulation; concurrently, IBM superconducting quantum hardware is utilized to evaluate the statistical behavior of the circuit family under physical noise. We introduce a dual-region image layout to resolve the nonce bootstrapping dependency, separating header recovery from payload recovery through independently derived keys. Experimental results confirm successful end-to-end message embedding and recovery on PNG images, demonstrating deterministic success under correct conditions and failure otherwise. The framework supports both text and image payloads; in the image-in-image configuration, a secret image is resized to a fixed resolution prior to embedding, enabling exact pixel-level recovery under correct contextual reconstruction.

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 $R^2$ of attention-based VQCs while using 40-50% fewer parameters, and consistently outperform equal-capacity MLPs (mean $R^2{=}0.829$ vs.\ MLP$_{720}$'s $0.753$ on Boston Housing, 3-seed average); \textbf{(ii)}FC-VQC's Type4 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${\approx}\,3$, 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 train poorly on chaotic forecasting, usually blamed on barren plateaus (exponentially vanishing gradients). Using an exactly simulable four-qubit variational quantum physics-informed circuit fit to Lorenz-63, we show the barren-plateau explanation fails: the failure is an architectural capacity ceiling fixed by the circuit time-encoding, not its trainable depth. Four measurements support this. (i) A McClean-comparable gradient-variance estimator sits at the local-cost Haar/2-design scale 2^(-2n)=3.9e-3 at n=4; on structurally live parameters it decays about ninefold with depth then saturates there, large enough to train, not an exponential collapse. (ii) At a common budget of 200 optimiser iterations (600, in three stages, for layer-wise), gradient descent, layer-wise, and SPSA reach the same order of magnitude of loss, so no optimiser unlocks a better basin. (iii) The output-Jacobian rank saturates at 33 from five layers on, so depth buys no new output directions. (iv) A Fourier analysis explains why: the qubit-1 phase encoding acts on the initial |0> and is inert, so the maximum accessible frequency is 2.5/t_max=0.83 Hz, identical at every depth and about 4.4x below the narrowest Lorenz component bandwidth. The corrected band has dimension 1+2x5=11 per observable, and 3x11=33 equals the measured rank ceiling exactly, unifying the two diagnostics. A trained depth sweep agrees: mean loss improves with depth then flattens once the rank saturates. We correct our earlier preprint diagnosis, which compared unnormalised gradient norms to the McClean threshold, and place the advantage of fixed reservoirs and classical echo-state networks in architecture, not quantum mechanics.

Barren-plateau results have established exponential gradient suppression as a widely cited obstacle to the scalability of variational quantum algorithms. When and whether these results extend to a given objective has been addressed through loss-specific arguments, but a general structural characterization has remained open. We show that the objective itself admits a fixed-observable representation if and only if the loss is affine in the measured statistics, thereby identifying the exact boundary of the standard concentration-based proof template. Existing transfer results for non-affine losses achieve this reduction under additional assumptions; our characterization implies that such a reduction is not structurally available for a class of non-affine objectives, placing them outside the automatic reach of the existing proof template. Beyond the affine regime, a chain-rule decomposition reveals three governing factors -- model responsivity, loss-side signal, and transmittance -- and induces a loss-class dichotomy: bounded-gradient losses inherit suppression, while amplification-capable losses can in principle counteract it. In the exponentially wide setting, both classes fail, but for different structural reasons. When the interface is instead designed at polynomial width -- exposing coarse-grained statistics rather than individual bitstring probabilities -- the exponential-dimensional obstruction is relaxed and the dichotomy plays a genuine role. In a numerical demonstration on a charge-conserving quantum system, the amplification-capable objective produces resolved gradients several orders of magnitude larger than affine and inheriting baselines at comparable shot budgets. Over the tested interval, its scaling trend is statistically distinguished from the exponential trend of both alternatives. The boundary is affine; what lies beyond it is a representation-design problem.

Quantum generative modeling is a rapidly evolving discipline at the intersection of quantum computing and machine learning. Contemporary quantum machine learning is generally limited to toy examples or heavily restricted datasets with few elements. This is not only due to the current limitations of available quantum hardware but also due to the absence of inductive biases arising from application-agnostic designs. Current quantum solutions must resort to tricks to scale down high-resolution images, such as relying heavily on dimensionality reduction or utilizing multiple quantum models for low-resolution image patches. Building on recent developments in classical image loading to quantum computers, we circumvent these limitations and train quantum Wasserstein GANs on the established classical MNIST and Fashion-MNIST datasets. Using the complete datasets, our system generates full-resolution images across all ten classes and establishes a new state-of-the-art performance with a single end-to-end quantum generator without tricks. As a proof-of-principle, we also demonstrate that our approach can be extended to color images, exemplified on the Street View House Numbers dataset. We analyze how the choice of variational circuit architecture introduces inductive biases, which crucially unlock this performance. Furthermore, enhanced noise input techniques enable highly diverse image generation while maintaining quality. Finally, we show promising results even under quantum shot noise conditions.

Quantum circuit design is a key bottleneck for practical quantum machine learning on complex, real-world data. We present an automated framework that discovers and refines variational quantum circuits (VQCs) using graph-based Bayesian optimization with a graph neural network (GNN) surrogate. Circuits are represented as graphs and mutated and selected via an expected improvement acquisition function informed by surrogate uncertainty with Monte Carlo dropout. Candidate circuits are evaluated with a hybrid quantum-classical variational classifier on the next generation firewall telemetry and network internet of things (NF-ToN-IoT-V2) cybersecurity dataset, after feature selection and scaling for quantum embedding. We benchmark our pipeline against an MLP-based surrogate, random search, and greedy GNN selection. The GNN-guided optimizer consistently finds circuits with lower complexity and competitive or superior classification accuracy compared to all baselines. Robustness is assessed via a noise study across standard quantum noise channels, including amplitude damping, phase damping, thermal relaxation, depolarizing, and readout bit flip noise. The implementation is fully reproducible, with time benchmarking and export of best found circuits, providing a scalable and interpretable route to automated quantum circuit discovery.

Variational quantum circuits (VQCs) are central to quantum machine learning, while recent progress in Kolmogorov-Arnold networks (KANs) highlights the power of learnable activation functions. We unify these directions by introducing the quantum variational activation function (QVAF), a general framework in which parameterized quantum circuits serve as learnable activation functions; in this work we study an efficient single-qubit instantiation called DatA Re-Uploading ActivatioN (DARUAN). We show that DARUAN with trainable data-preprocessing weights can realize an exponentially growing accessible frequency support with the number of re-uploading repetitions; for an explicit geometric choice of these weights, this gives a capacity-level exponential parameter reduction relative to independently parameterized Fourier activations. Embedding DARUAN into KAN yields the quantum-inspired Kolmogorov-Arnold Network (QKAN), which retains the interpretability of the KAN architecture while improving parameter efficiency, expressivity, and generalization. We further introduce layer extension and the hybrid QKAN (HQKAN) architecture to improve scalability and computational efficiency, enabling QKAN modules to act as compact replacements for multi-layer perceptrons (MLPs) in large-scale models. We provide theoretical analysis and extensive experiments on function regression, image classification, and autoregressive generative language modeling, demonstrating the efficiency and scalability of QKANs. Because the single-qubit circuits are efficiently simulable on classical quantum simulators, QKANs have quantum-inspired advantage in parameter efficiency and training stability; DARUANs and QKANs serve as present-day validation of the QVAF concept, and the trained DARUANs are directly executable and feasible on current noisy intermediate-scale quantum (NISQ) hardware for inference validation.

Conic programs arising in physics, quantum information, machine learning, and engineering are often defined over sparse graphs. Although such problems can be solved in polynomial time using classical interior-point solvers, the computational complexity scales unfavorably with graph size. We propose a variational quantum paradigm for solving conic programs, including quadratically constrained quadratic programs and semidefinite programs. We encode primal variables via the state of a parameterized quantum circuit (PQC) and dual variables via the probability mass function associated with a second PQC. The Lagrangian function can thus be expressed as scaled expectations of quantum observables. We pursue approximately stationary points of the Lagrangian by minimizing/maximizing the Lagrangian over the parameters of the first/second PQC. This is accomplished in a hybrid fashion: gradients are estimated using the two PQCs, while their parameters are updated classically using a primal-dual method. We propose permuting primal variables so that related observables have a banded form, enabling efficient measurement. We provide a complexity analysis that is useful to determine which problem types may enjoy quantum advantage. The framework is applied to the AC OPF problem, a large-scale optimization problem central to electric power system operation. Numerical tests on the IEEE 57-node system using PennyLane's simulator show that the proposed doubly variational quantum framework can find high-quality OPF solutions. While this demonstration does not yield a quantum speedup, the results serve as a proof-of-concept and highlight challenges toward practical quantum advantage. Although showcased for OPF, the framework has broader scope, including conic programs with many variables and constraints, problems defined over sparse graphs, and training quantum machine learning models to satisfy constraints.

In this work, a novel algorithm structured on Lie algebras for the approximate quantum state preparation problem is proposed, addressing a challenge of fundamental importance in many areas of quantum computing. The algorithm uses a variational quantum circuit designed on the Standard Recursive Block Basis (SRBB), a hierarchical construction for the matrix algebra of the $SU(2^n)$ group, which is capable of linking the variational parameters with the topology of the Lie group. Compared to the full algebra, using only diagonal components reduces the number of CNOTs by an exponential factor, as well as the circuit depth, in full agreement with the relaxation principle inherent to the approximation methodology of minimizing resources while achieving high accuracy. The desired quantum state is then approximated by a novel quantum neural network, which is designed based on the diagonal SRBB sub-algebra. This approach provides a new scheme for approximate quantum state preparation in a variational framework and a specific use case for the SRBB hierarchy. The performance of the algorithm is assessed with different loss functions, such as fidelity, trace distance, and Frobenius norm, in relation to two optimizers: Adam and Nelder-Mead. The results highlight the potential of SRBB in close connection with the geometry of unitary groups, achieving high accuracy of up to 4 qubits in simulation, but also its current limitations with an increasing number of qubits. Additionally, the approximate SRBB-based QSP algorithm has been tested on real quantum devices to assess its performance with a small number of qubits.

Quantum neural networks (QNNs) have attracted growing interest for scientific machine learning, yet in regression settings they often suffer from limited trainability under noisy gradients and ill-conditioned optimization. We propose a hybrid quantum--classical regression framework designed to mitigate these bottlenecks. Our model prepends a lightweight classical embedding that acts as a learnable geometric preconditioner, reshaping the input representation to better condition a downstream variational quantum circuit. Building on this architecture, we introduce a curriculum optimization protocol that progressively increases circuit depth and transitions from SPSA-based stochastic exploration to Adam-based gradient fine-tuning. We evaluate the approach on PDE-informed regression benchmarks and standard regression datasets under a fixed training budget in a simulator setting. Empirically, the proposed framework consistently improves over pure QNN baselines and yields more stable convergence in data-limited regimes. We further observe reduced structured errors that are visually correlated with oscillatory components on several scientific benchmarks, suggesting that geometric preconditioning combined with curriculum training is a practical approach for stabilizing quantum regression.