PhasorFlow: A Python Library for Unit Circle Based Computing

Abstract

We present PhasorFlow, an open-source Python library for computing on the $S^1$ unit circle. Inputs are encoded as complex phasors $z=e^{iϕ}$ on the $N$-torus ($\mathbb{T}^N$); as computation proceeds through unitary wave-interference gates, global norm is preserved while components drift into $\mathbb{C}^N$, letting algorithms leverage continuous geometric gradients. PhasorFlow makes three contributions. First, we formalize the Phasor Circuit model ($N$ threads, $M$ gates) with a 22-gate library spanning standard-unitary, non-linear, neuromorphic, and encoding operations under full matrix-algebra simulation. Second, we present the Variational Phasor Circuit (VPC), analogous to variational quantum circuits, optimizing continuous phase parameters for classification. Third, we introduce the Phasor Transformer, replacing $QK^TV$ attention with a parameter-free DFT token-mixing layer inspired by FNet. We validate on spatial classification, time-series prediction, financial volatility, neuromorphic tasks, and -- for the VPC -- real motor-imagery EEG, where it matches standard baselines at a fraction of their parameters. We characterize the models honestly: the VPC is a parameter-efficient phase-linear classifier with a parity ceiling that depth cannot raise, and the Phasor Transformer benefits from depth before saturating, competitive but not superior. This positions unit-circle computing as a deterministic, lightweight paradigm on classical hardware. Available at https://github.com/mindverse-computing/phasorflow.