# LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits

arXiv: https://arxiv.org/abs/2204.11787 · HTML: https://mlsanity.com/paper/2204.11787/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Apr 2022.

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

- [A Quantum Natural Language Processing Approach to Pronoun Resolution](https://mlsanity.com/paper/2208.05393/) (2022)
- [Photonic Quantum Computers](https://mlsanity.com/paper/2409.08229/) (2024)

## Abstract (arXiv)

We introduce the LOv-calculus, a graphical language for reasoning about linear optical quantum circuits with so-called vacuum state auxiliary inputs. We present the axiomatics of the language and prove its soundness and completeness: two LOv-circuits represent the same quantum process if and only if one can be transformed into the other with the rules of the LOv-calculus. We give a confluent and terminating rewrite system to rewrite any polarisation-preserving LOv-circuit into a unique triangular normal form, inspired by the universal decomposition of Reck et al. (1994) for linear optical quantum circuits.