Graph States, Pivot Minor, and Universality of (X,Z)-measurements

TL;DR

This paper demonstrates that any graph can be derived as a pivot-minor of a planar or triangular grid graph and establishes that (X,Z)-measurements on triangular grid graph states form a universal quantum computation model.

Contribution

It proves the universality of (X,Z)-measurements on triangular grid graph states and links graph minor theory with quantum measurement-based computation.

Findings

Any graph is a pivot-minor of a planar graph.

Triangular grid graph states with (X,Z)-measurements are universal for quantum computation.

The results connect graph theory with quantum information processing.

Abstract

The graph state formalism offers strong connections between quantum information processing and graph theory. Exploring these connections, first we show that any graph is a pivot-minor of a planar graph, and even a pivot minor of a triangular grid. Then, we prove that the application of measurements in the (X,Z)-plane over graph states represented by triangular grids is a universal measurement-based model of quantum computation. These two results are in fact two sides of the same coin, the proof of which is a combination of graph theoretical and quantum information techniques.