PhasorFlow: A Python Library for Unit Circle Based Computing

We present the Variational Phasor Circuit (VPC), a deterministic classical learning architecture on the continuous $S^1$ unit-circle manifold. Inspired by variational quantum circuits, VPC replaces dense weight matrices with trainable phase shifts, local unitary mixing, and structured interference in the ambient complex space, giving a unified method for binary and multi-class classification of spatially distributed signals. We evaluate VPC on real motor-imagery electroencephalography (EEG) from the PhysioNet Motor Movement/Imagery database (10 subjects, Common Spatial Pattern features, subject-wise cross-validation), where it attains a mean decoding accuracy of $0.60$ -- the highest among standard brain--computer-interface baselines (linear discriminant analysis, logistic regression, RBF-SVM, and a multilayer perceptron) -- using an order of magnitude fewer parameters and the lowest cross-subject variance. We also characterize capacity honestly: with phase-only shifts and unitary mixing, VPC realizes a linear decision function in a fixed cosine/sine feature lifting, well matched to the largely separable band-power structure of EEG but unable to represent parity-type functions, a ceiling that depth does not raise. These results position unit-circle phase interference as a parameter-efficient alternative to dense neural computation for signal classification, and motivate VPC both as a standalone classifier and a front-end for hybrid phasor-quantum systems.

Transformer models have redefined sequence learning, yet dot-product self-attention introduces a quadratic token-mixing bottleneck for long-context time-series. We introduce the Phasor Transformer block, a phase-native alternative representing sequence states on the unit-circle manifold $S^1$. Each block combines lightweight trainable phase-shifts with parameter-free Discrete Fourier Transform (DFT) token coupling, achieving global $\mathcal{O}(N\log N)$ mixing without explicit attention maps. Stacking these blocks defines the Large Phasor Model (LPM). We validate LPM on autoregressive time-series prediction over synthetic multi-frequency benchmarks against honest baselines: it beats a zero-parameter persistence baseline and, with the corrected gradient path, improves monotonically with depth before saturating, while remaining competitive-but-not-superior to self-attention at a fraction of the parameter count. Our results establish an explicit efficiency--accuracy frontier, showing that scalable temporal modeling in oscillatory domains can emerge from geometry-constrained phase computation with deterministic global coupling.

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.