''It’s the Symmetry, Stupid!'' Gearing Up Tensor Networks for the Topological Quantum Revolution

Introduction

This proposal is set in the era of the second quantum revolution, in which there is a strong need for a computational framework to describe and simulate topological materials and quantum devices possessing exotic particle statistics.

Key Actors

• The two leading actors: symmetries and tensor networks.

The Story

• The story: while the topological materials seem to defy Landau’s symmetry breaking paradigm as they do not exhibit a local order parameter, tensor networks strike back with the emergence of a local “pulling-through” symmetry of the local tensors describing them.

The Catch

• The catch: the symmetry happens in the shadow world of entanglement degrees of freedom, is in general non-invertible, and speaks the language of bimodule categories.

The Magic

• The magic: the tensor network picture makes the abstract concepts and mathematical tools needed to describe topological and critical systems, such as cohomology theory and bimodule categories, very tangible and concrete as wavefunctions of quantum spin systems which can easily be put on a computer.

The Mission

• The mission:
  1. Develop the mathematics of tensor network representations of (higher) categorical and non-invertible symmetries.
  2. Construct a computational toolbox which exploits those symmetries.
  3. Construct explicit matrix product operator intertwiners for dualities between theories with non-Abelian / categorical symmetries.
  4. Construct novel CFTs from categorical data.
  5. Create a bridge between bimodule categories and integrability.

The Crescendo

• The crescendo: tensor networks with categorical symmetries take over the world by
  1. Setting up real-space renormalization group flows in which unwanted relevant perturbations are crushed.
  2. Constructing and evaluating quantum error correcting codes based on non-Abelian anyons.

The Take-Away

• The take-away: deep theoretical ideas and state of the art computational aspects are not parallel lines of development, but are one and the same and must be fused with each other.

The Moral

• The moral: For things to remain the same, everything must change.