PhasorFlow: A Python Library for Unit Circle Based Computing

TL;DR

PhasorFlow introduces a Python library for unit circle based computing using complex phasors, enabling continuous geometric gradient algorithms for machine learning and neuromorphic tasks on classical hardware.

Contribution

It formalizes the Phasor Circuit model, presents a variational circuit for optimization, and introduces a DFT-based token mixing layer as a lightweight alternative to quantum attention.

Findings

Validated on non-linear classification and time-series prediction.

Demonstrated effectiveness in financial volatility detection.

Showcased neuromorphic applications like neural binding and memory.

Abstract

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 $z = e^{i\theta}$ 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.