Quantum Kernels for Parity-Structured Classification: A Hybrid Pipeline

Parity (XOR) classification requires detecting discrete, high-order feature interactions that smooth classical kernels cannot efficiently capture. We study how quantum kernel advantage depends on parity complexity, the number of features entering the XOR rule, and find a clear threshold behavior. We pair a ZZ quantum feature map with binary {0, pi} encoding (features median thresholded before circuit input) to expose parity structure. A binary encoding ablation, RBF SVM trained on the identical {0, pi} features, separates encoding from circuit effects: at low complexity (n = 5 features), binary RBF achieves 83.4% +/- 1.7% and the quantum kernel 81.2% +/- 1.9%, showing encoding drives performance there. At high complexity (n = 11 features, 11 qubits, r = 3 ZZ repetitions), all classical methods collapse to near-random (approx. 50%), binary RBF reaches only 54.3% +/- 1.1%, and the quantum ZZ kernel achieves 66.3% +/- 3.2% (mean +/- std, 10 seeds), a +12.0 percentage-point margin over the binary ablation and approx. 7x higher kernel-target alignment (0.094 +/- 0.020 vs. 0.013 +/- 0.001). These results identify parity complexity as a concrete axis along which genuine quantum kernel advantage, not attributable to encoding alone, emerges.

Fixed-Reservoir vs Variational Quantum Architectures for Chaotic Dynamics: Benchmarking QRC and QPINN on the Lorenz System

Deploying quantum machine learning on NISQ devices requires architectures where training overhead does not negate computational advantages. We systematically compare two quantum approaches for chaotic time-series prediction on the Lorenz system: a variational Quantum Physics-Informed Neural Network (QPINN) and a Quantum Reservoir Computing (QRC) framework utilizing a fixed transverse-field Ising Hamiltonian. Under matched resources ($4$--$5$ qubits, $2$--$3$ layers), QRC achieves an $81\%$ lower mean-squared error (test MSE $3.2 \pm 0.6$ vs. $47.9 \pm 36.6$ for QPINN) while training $\sim 52,000\times$ faster ($0.2$\,s vs. $\sim 2.4$\,h per seed). Drawing on the classical delay-embedding principle, we formalize a temporal windowing technique within the QRC pipeline that improves attractor reconstruction by providing bounded, structured input history. Analysis reveals that QPINN instability stems from capacity limitations and competing loss terms rather than barren plateaus; gradient norms remained large ($10^3$--$10^4$), ruling out exponential suppression at this scale. These failure modes are absent by construction in the non-variational QRC approach. We validate robustness across three canonical systems (Lorenz, Rössler, and Lorenz-96), where QRC consistently achieves low test MSE ($3.1 \pm 0.6$, $1.8 \pm 0.1$, and $12.4 \pm 0.6$, respectively) with sub-second training. Our findings suggest the fixed-reservoir architecture is a primary driver of QRC's advantage at these scales, warranting further investigation at larger qubit counts and on hardware where quantum-specific advantages are expected to emerge.

Adaptive Graph of Thoughts: Test-Time Adaptive Reasoning Unifying Chain, Tree, and Graph Structures

Large Language Models (LLMs) have demonstrated impressive reasoning capabilities, yet their performance is highly dependent on the prompting strategy and model scale. While reinforcement learning and fine-tuning have been deployed to boost reasoning, these approaches incur substantial computational and data overhead. In this work, we introduce Adaptive Graph of Thoughts (AGoT), a dynamic, graph-based inference framework that enhances LLM reasoning solely at test time. Rather than relying on fixed-step methods like Chain of Thought (CoT) or Tree of Thoughts (ToT), AGoT recursively decomposes complex queries into structured subproblems, forming an dynamic directed acyclic graph (DAG) of interdependent reasoning steps. By selectively expanding only those subproblems that require further analysis, AGoT unifies the strengths of chain, tree, and graph paradigms into a cohesive framework that allocates computation where it is most needed. We validate our approach on diverse benchmarks spanning multi-hop retrieval, scientific reasoning, and mathematical problem-solving, achieving up to 46.2% improvement on scientific reasoning tasks (GPQA) - comparable to gains achieved through computationally intensive reinforcement learning approaches and outperforming state-of-the-art iterative approaches. These results suggest that dynamic decomposition and structured recursion offer a scalable, cost-effective alternative to post-training modifications, paving the way for more robust, general-purpose reasoning in LLMs.

Topological Understanding of Neural Networks, a survey

We look at the internal structure of neural networks which is usually treated as a black box. The easiest and the most comprehensible thing to do is to look at a binary classification and try to understand the approach a neural network takes. We review the significance of different activation functions, types of network architectures associated to them, and some empirical data. We find some interesting observations and a possibility to build upon the ideas to verify the process for real datasets. We suggest some possible experiments to look forward to in three different directions.