The GHZ/W-calculus contains rational arithmetic

Abstract

Graphical calculi for representing interacting quantum systems serve a number of purposes: compositionally, intuitive graphical reasoning, and a logical underpinning for automation. The power of these calculi stems from the fact that they embody generalized symmetries of the structure of quantum operations, which, for example, stretch well beyond the Choi-Jamiolkowski isomorphism. One such calculus takes the GHZ and W states as its basic generators. Here we show that this language allows one to encode standard rational calculus, with the GHZ state as multiplication, the W state as addition, the Pauli X gate as multiplicative inversion, and the Pauli Z gate as additive inversion.

In Proceedings HPC 2010, arXiv:1103.2268

References in corpus (4)

• Three qubits can be entangled in two inequivalent ways
• Quantum Picturalism
• Phase groups and the origin of non-locality for qubits
• Open Graphs and Monoidal Theories

Cited by in corpus (13)

• Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing
• The algebra of entanglement and the geometry of composition
• ZX-calculus for the working quantum computer scientist
• Completeness for arbitrary finite dimensions of ZXW-calculus, a unifying calculus
• Reasoning with !-Graphs
• Light-Matter Interaction in the ZXW Calculus
• Differentiating and Integrating ZX Diagrams with Applications to Quantum Machine Learning
• Kindergarden quantum mechanics graduates (...or how I learned to stop gluing LEGO together and love the ZX-calculus)
• Towards Normal Forms for GHZ/W Calculus
• ZX-calculus is Complete for Finite-Dimensional Hilbert Spaces
• Graphical Calculi and their Conjecture Synthesis
• (Modular) Effect Algebras are Equivalent to (Frobenius) Antispecial Algebras
• A Diagrammatic Axiomatisation for Qubit Entanglement