The ZX-calculus is complete for stabilizer quantum mechanics

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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.
Original languageEnglish
Article number093021
Number of pages30
JournalNew Journal of Physics
Issue number9
Publication statusPublished - 17 Sept 2014


  • quantum foundations
  • quantum computing
  • logic
  • graphical calculus
  • stabilizer quantum mechanics
  • graph states


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