Dibakar Sigdel

Dibakar Sigdel, Namuna Panday

We present PhasorFlow, an open-source Python library introducing a computational paradigm operating on the $S^1$ unit circle. Inputs are encoded as complex phasors $z = e^{iθ}$ on the $N$-Torus ($\mathbb{T}^N$). As computation proceeds via unitary wave interference gates, global norm is preserved while individual components drift into $\mathbb{C}^N$, allowing algorithms to natively leverage continuous geometric gradients for predictive learning. PhasorFlow provides three core contributions. First, we formalize the Phasor Circuit model ($N$ unit circle threads, $M$ gates) and introduce a 22-gate library covering Standard Unitary, Non-Linear, Neuromorphic, and Encoding operations with full matrix algebra simulation. Second, we present the Variational Phasor Circuit (VPC), analogous to Variational Quantum Circuits (VQC), enabling optimization of continuous phase parameters for classical machine learning tasks. Third, we introduce the Phasor Transformer, replacing expensive $QK^TV$ attention with a parameter-free, DFT-based token mixing layer inspired by FNet. We validate PhasorFlow on non-linear spatial classification, time-series prediction, financial volatility detection, and neuromorphic tasks including neural binding and oscillatory associative memory. Our results establish unit circle computing as a deterministic, lightweight, and mathematically principled alternative to classical neural networks and quantum circuits. It operates on classical hardware while sharing quantum mechanics' unitary foundations. PhasorFlow is available at https://github.com/mindverse-computing/phasorflow.

Dibakar Sigdel

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 \textbf{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 \textbf{Large Phasor Model (LPM)}. We validate LPM on autoregressive time-series prediction over synthetic multi-frequency benchmarks. Operating with a highly compact parameter budget, LPM learns stable global dynamics and achieves competitive forecasting behavior compared to conventional self-attention baselines. Our results establish an explicit efficiency-performance frontier, demonstrating that large-model scaling for time-series can emerge from geometry-constrained phase computation with deterministic global coupling, offering a practical path toward scalable temporal modeling in oscillatory domains.

Dibakar Sigdel

We present the \textbf{Variational Phasor Circuit (VPC)}, a deterministic classical learning architecture operating on the continuous $S^1$ unit circle manifold. Inspired by variational quantum circuits, VPC replaces dense real-valued weight matrices with trainable phase shifts, local unitary mixing, and structured interference in the ambient complex space. This phase-native design provides a unified method for both binary and multi-class classification of spatially distributed signals. A single VPC block supports compact phase-based decision boundaries, while stacked VPC compositions extend the model to deeper circuits through inter-block pull-back normalization. Using synthetic brain-computer interface benchmarks, we show that VPC can decode difficult mental-state classification tasks with competitive accuracy and substantially fewer trainable parameters than standard Euclidean baselines. These results position unit-circle phase interference as a practical and mathematically principled alternative to dense neural computation, and motivate VPC as both a standalone classifier and a front-end encoding layer for future hybrid phasor-quantum systems.