# PhasorFlow: A Python Library for Unit Circle Based Computing

arXiv: https://arxiv.org/abs/2603.15886 · HTML: https://mlsanity.com/paper/2603.15886/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 37 references that are in the map.
- First posted on arXiv 16 Mar 2026.

## Builds on (most cited first, 9 of 9)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Variational Quantum Algorithms](https://mlsanity.com/paper/2012.09265/) (2020)
- [Hopfield Networks is All You Need](https://mlsanity.com/paper/2008.02217/) (2020)
- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [Parameterized quantum circuits as machine learning models](https://mlsanity.com/paper/1906.07682/) (2019)
- [Unitary Evolution Recurrent Neural Networks](https://mlsanity.com/paper/1511.06464/) (2015)
- [Deep Complex Networks](https://mlsanity.com/paper/1705.09792/) (2017)
- [Full-Capacity Unitary Recurrent Neural Networks](https://mlsanity.com/paper/1611.00035/) (2016)
- [Computing on Functions Using Randomized Vector Representations](https://mlsanity.com/paper/2109.03429/) (2021)

## Built on by (most cited first, 2 of 2)

- [The Phasor Transformer: Resolving Attention Bottlenecks on the Unit Circle](https://mlsanity.com/paper/2603.17433/) (2026)
- [Variational Phasor Circuits for Phase-Native Brain-Computer Interface Classification](https://mlsanity.com/paper/2603.18078/) (2026)

## Abstract (arXiv)

We present PhasorFlow, an open-source Python library for computing on the S1 unit circle. Inputs are encoded as complex phasors z=eiφ on the N-torus (mathbbTN); as computation proceeds through unitary wave-interference gates, global norm is preserved while components drift into mathbbCN, 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 introduce the Variational Phasor Circuit (VPC), a trainable phase-native classifier analogous to variational quantum circuits. Third, we introduce the Phasor Transformer block and Large Phasor Model (LPM), replacing QKTV attention with a parameter-free DFT token-mixing layer. We validate the framework on financial volatility detection, neuromorphic associative memory, neural binding, period finding, and algorithmic logic applications that are unique to the library. This positions unit-circle computing as a deterministic, lightweight paradigm on classical hardware. Available at https://github.com/mindverse-computing/phasorflow.