Section: New Results

Quantum Computing

Participants : Simon Perdrix, Quanlong Wang.

• ZX-calculus

  The ZX-calculus is a powerful diagrammatic language for quantum mechanics and quantum information processing. The completeness of the ZX-calculus is crucial: the language would be complete if any equation involving two diagrams representing the same quantum evolution can be derived using the rules of the language. While the language is known to be incomplete in general with no obvious way to add some new rules [75], two interesting fragments have been studied: the $\pi /2$ and the $\pi /4$-fragments, obtained by restricting the angles of diagrams to be multiples of $\pi /2$ and $\pi /4$ respectively.

  The $\pi /4$-fragment is approximatively universal for quantum mechanics, i.e. any quantum evolution can be approximated with an arbitrary accuracy using a diagram involving only angles multiple of $\pi /4$. The completeness of this fragment was one of the main open question in this domain. We have proved that this fragment is incomplete. We exhibit a fairly simple equation called supplementarity and we prove that this equation cannot be derived in the ZX-calculus. We propose as a consequence, to add supplementarity to the set of rules of the ZX-calculus. This result has been published at MFCS 2016 [20].

  The $\pi /2$-fragment is not universal, even approximatively. However it corresponds to the so-called stabiliser quantum mechanics, an interesting fragment of quantum mechanics. The $pi/2$-fragment is known to be complete for stabiliser quantum mechanics [33]. We have proved recently that the rules of the language can be simplified, leading to a simpler set of axioms. Moreover we have proved that most of the remaining rules being necessary are the completeness of the $\pi /2$-fragment. This result has been published at QPL 2016 [16].

• Causal Graph Dynamics

  Causal Graph Dynamics [30] extend Cellular Automata to arbitrary, bounded-degree, time-varying graphs. The whole graph evolves in discrete time steps, and this global evolution is required to have a number of physics-like symmetries: shift-invariance (it acts everywhere the same) and causality (information has a bounded speed of propagation). We add a further physics-like symmetry, namely reversibility. This result has been presented at RC 2016 [15].