LOv-Calculus: A Graphical Language for Linear Optical Quantum Circuits
Alexandre Cl\'ement, Nicolas Heurtel, Shane Mansfield, Simon Perdrix,, Beno\^it Valiron
TL;DR
The paper introduces LOv-calculus, a graphical language for linear optical quantum circuits, providing axioms, soundness, completeness, and a rewrite system for circuit normalization.
Contribution
It presents the first complete and sound graphical calculus for linear optical quantum circuits with auxiliary vacuum inputs.
Findings
Proves soundness and completeness of LOv-calculus.
Develops a confluent rewrite system for circuit normalization.
Provides a unique normal form inspired by Reck's decomposition.
Abstract
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.
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