Graphical Symplectic Algebra

TL;DR

This paper develops a unified graphical language for affine Lagrangian relations and stabiliser quantum circuits, enabling concise reasoning about complex systems in classical mechanics and quantum information.

Contribution

It introduces a complete graphical presentation for affine Lagrangian and coisotropic relations, unifying classical and quantum systems through scalable graph-based notation.

Findings

Graphical languages for classical and quantum systems are unified.

Scalable notation simplifies reasoning about composite systems.

Impedance matrices are represented similarly to quantum graph states.

Abstract

We give complete presentations for the dagger-compact props of affine Lagrangian and coisotropic relations over an arbitrary field. This provides a unified family of graphical languages for both affinely constrained classical mechanical systems, as well as odd-prime-dimensional stabiliser quantum circuits. To this end, we present affine Lagrangian relations by a particular class of undirected coloured graphs. In order to reason about composite systems, we introduce a powerful scalable notation where the vertices of these graphs are themselves coloured by graphs. In the setting of stabiliser quantum mechanics, this scalable notation gives an extremely concise description of graph states, which can be composed via ``phased spider fusion.'' Likewise, in the classical mechanical setting of electrical circuits, we show that impedance matrices for reciprocal networks are presented in essentially the same way.