PIRSA:14120040

Video URL

https://pirsa.org/14120040

Completeness Results for Graphical Quantum Process Languages

Backens, M. (2014). Completeness Results for Graphical Quantum Process Languages. Perimeter Institute for Theoretical Physics. https://pirsa.org/14120040

Backens, Miriam. Completeness Results for Graphical Quantum Process Languages. Perimeter Institute for Theoretical Physics, Dec. 10, 2014, https://pirsa.org/14120040 

          @misc{ scivideos_PIRSA:14120040,
            doi = {10.48660/14120040},
            url = {https://pirsa.org/14120040},
            author = {Backens, Miriam},
            keywords = {Quantum Foundations, Quantum Information},
            language = {en},
            title = {Completeness Results for Graphical Quantum Process Languages},
            publisher = {Perimeter Institute for Theoretical Physics},
            year = {2014},
            month = {dec},
            note = {PIRSA:14120040 see, \url{https://scivideos.org/index.php/pirsa/14120040}}
          }
Talk numberPIRSA:14120040
DOI10.48660/14120040
Source RepositoryPIRSA
Collection
Talk Type Scientific Series
Subject

Abstract

From Feynman diagrams via Penrose graphical notation to quantum circuits, graphical languages are widely used in quantum theory and other areas of theoretical physics. The category-theoretical approach to quantum mechanics yields a new set of graphical languages, which allow rigorous pictorial reasoning about quantum systems and processes. One such language is the ZX-calculus, which is built up of elements corresponding to maps in the computational and the Hadamard basis. This calculus is universal for pure state qubit quantum mechanics, meaning any pure state, unitary operation, and post-selected pure projective measurement can be represented. It is also sound, meaning any graphical rewrite corresponds to a valid equality when translated into matrices.

While the calculus is not complete for general quantum mechanics, I show that it is complete for stabilizer quantum mechanics and for the single-qubit Clifford+T group. This means that within those subtheories, any equality that can be derived using matrices can also be derived graphically.

The ZX-calculus can thus be applied to a wide range of problems in quantum information and quantum foundations, from the analysis of quantum non-locality to the verification of measurement-based quantum computation and error-correcting codes. I also show how to construct a ZX-like graphical calculus for Spekkens' toy bit theory and give its associated completeness proof.