The ZX−calculus is complete for stabilizer quantum mechanics

Journal article

The ZX-calculus is a graphical calculus for reasoning about quantum systems and processes. It is known to be universal for pure state qubit quantum mechanics (QM), meaning any pure state, unitary operation and post-selected pure projective measurement can be expressed in the ZX-calculus. The calculus is also sound, i.e. any equality that can be derived graphically can also be derived using matrix mechanics. Here, we show that the ZX-calculus is complete for pure qubit stabilizer QM, meaning any equality that can be derived using matrices can also be derived pictorially. The proof relies on bringing diagrams into a normal form based on graph states and local Clifford operations.

Authors: Backens, M

• Journal: New Journal of Physics
• Volume: 16
• Issue: 9
• Pages: 093021
• Publication date: 2014-09-01
• DOI: 10.1088/1367-2630/16/9/093021
• ISSN: 1367-2630