Purdue Quantum Theory Seminar

We construct a local decoder for the 2D toric code using ideas from the hierarchical classical cellular automata of Tsirelson and Gács. Our decoder is a circuit of strictly local quantum operations preserving a logical state for exponential time in the presence of circuit-level noise without the need for non-local classical computation or communication. Our construction is not translation invariant in spacetime but can be made time-translation invariant in 3D with stacks of 2D toric codes. This solves the open problem of constructing a local topological quantum memory below four dimensions.

This talk is based on joint work [arXiv:2412.19803] with Shankar Balasubramanian and Ethan Lake.

In this talk, I will explain the concept of SPT-Sewing, which is a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible \(G \times \mathrm{Rep}(G) \times G\) symmetry and invertible domain walls in the quantum double associated with the group \(G\). We prove this correspondence when \(G\) is Abelian and provide evidence for the general case by studying the quantum double model for \(G = S_3\). We also use our method to construct anchoring domain walls, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.

I will report on arXiv:2501.0528 where we consider the following decision problem: on input \(G\in \mathcal D\) and \(H\in \mathcal T\), decide if there is a surjective homomorphism from \(G\) onto \(H\).

We prove that the problem is NP-complete when \(\mathcal D\) is the class of all finitely presented groups and \(\mathcal T\) is the class of direct products \(\mathbb Z^d\times Q\) where \(Q\) is a finite group, the class of virtually cyclic groups, or the class of a single fixed dihedral group that is not nilpotent.

Work of Kuperberg and Samperton previously showed the problem is NP-complete for target a single non-abelian simple group arXiv:1707.03811.

Our techniques involve reducing epimorphism to decision problems about equations over groups, which I will explain in the talk.

This is joint work with Jerry Shen (UTS) and Armin Weiß (Stuttgart).

The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.

In this talk, we will give an overview of the current status of the classification program for modular categories. We will also present a construction of non-group-theoretical modular categories of certain dimensions. If time allows, we will present some open questions and the relation of this result and the classification by rank.

From physics lore, topological phases in the presence of a symmetry have the structure of a G-crossed braided tensor category, where the objects are the defects of the symmetry. In this talk we will show how to arrive at this structure using operator algebras, starting from lattice systems in the presence of a symmetry. Along the way, we will give a general recipe to create a symmetry defect. Finally we will discuss the novel consequences of this approach. In particular, it gives us a way to calculate the symmetry fractionalization class in the bulk using local unitaries.

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Jacob Lurie's classification of fully extended topological field theories applies in a broad and abstract setting across all dimensions. However, concrete examples remain scarce. In three dimensions, the Turaev-Viro theory has long been conjectured to fit within this framework. In this talk, we will establish that it necessarily does, outline a sketch of the proof, and explore the essential role of skein theory in this context.