Richard East

Diagrams are ubiquitous in physics and have catalysed progress on numerous occasions. From tensor networks and quantum circuits to Feynman diagrams, there are few areas of physics that don’t employ some informal pictorial reasoning. These diagrams represent the underlying  mathematical operations and aid  physical interpretation, but cannot generally be computed with directly. In this thesis the ZXH-calculus, a graphical language based on the ZX-calculus, is offered as a prototype for a formal diagrammatic calculational tool for theoretical physics involving spin. This extends the ZXH-calculus (and more broadly the ZX family of calculi to which it belongs) beyond its traditional domain which has largely  been dominated by quantum computing. In order to do this  the spin lattices taken from condensed matter physics are studied. It is also shown how spin-networks of the form often seen as the state-space of loop quantum  gravity (LQG) can diagrammatised along with operators acting on them.

To achieve this a diagrammatic form of SU(2) representation theory is outlined. Following this  in condensed matter a number of results are shown. The 1D  AKLT state, a symmetry protected topological state, is expressed in the ZXH-calculus by  developing a representation of spins higher than 1/2 within  the calculus. By exploiting  the simplifying power of  the ZXH-calculus rules it is shown how this representation straightforwardly recovers the AKLT matrix-product state representation, the existence of topologically protected edge states, and the non-vanishing of a  string order parameter. Extending beyond these known properties, the diagrammatic approach also  allows one to analytically derive that the Berry phase of any finite-length 1D AKLT chain is π. In addition, an alternative proof that the 2D AKLT state on a hexagonal lattice can be reduced to a graph state, demonstrating that it is a universal quantum computing resource.  Continuing on the theme of condensed matter it is then shown  how one can build 2D higher-order topological phases diagrammatically, which is used to illustrate a symmetry-breaking phase transition.

Turning to LQG the first step is the analysis of  Yutsis diagrams, a standard graphical calculus  used  in quantum chemistry and quantum gravity,  which captures the  main features of SU(2) representation theory. Second, it  is shown  how  it embeds  within Penrose’s  binor calculus. The two are then subsumed and rewritten into  ZXH-diagrams. In the process  we show how the SU(2) invariance  of Wigner symbols is trivially provable in the ZXH-calculus. Additionally, we  show how we can explicitly diagrammatically calculate 3jm, 4jm and 6j symbols. It has long been  thought that quantum gravity should be closely aligned to quantum information theory. In this paper, we  present a way in which this connection can be made exact, by writing the spin-networks of loop quantum gravity in the ZX-diagrammatic language of quantum computation. Finally after outlining the motivation  for considering spin-networks as the quantisation of space, the geometric operators are discussed, and in specific cases diagrammatic versions of the operators are provided. More generally what is done here shows a route by which LQG can be interpreted in quantum informational terms by rewriting its kinematical states as networks of qubits in ZXH.

In total these results demonstrate that the ZXH-calculus is a powerful language for representing quantum systems and even  allows for the computation on physical states entirely graphically. Within condensed matter it is hoped this will pave the way to  develop more efficient many-body algorithms and giving a novel diagrammatic  perspective on quantum phase transitions. In LQG it is hoped this re-imagining of its state space will spur further integration of quantum information and gravity.