Classification of 2D Homological Stabilizer Codes
The discovery of quantum error correction and fault-tolerance were major theoretical breakthroughs on the road towards building a full-fledged quantum computer. Since then thresholds have increased and geometric constraints on the underlying architecture have been added. Homological stabilizer codes provide a method for constructing stabilizer codes constrained to a 2D plane. In this talk I will define and proceed to classify all 2D homological stabilizer codes. I will show that Kitaev’s toric code and the topological color codes arise naturally in this classification. I will finally show, up to a set of equivalence relations, that these are the only 2D homological stabilizer codes.
Speaker Details
My current research is in quantum error correction (QEC) and fault-tolerance (FT). More specifically, I focus on QEC & FT in low-dimensional quantum architectures. These systems include toric codes, color codes, Levin-Wen string-net models, as well as traditional stabilizer codes with local stabilizer generators. My main focus in this area has been to evaluate the feasibility of these systems for use as FT quantum computing architectures. With my advisor, Andrew Landahl, I have studied the color codes in depth. This study included calculating the fault-tolerance threshold, devising new decoding routines, and constructing architectures for quantum computing in these systems.
Through these studies I became interested in the topological properties of these systems such as their topological order and anyon characteristics. Leveraging results from graph theory, I set out to classify the topological qubit stabilizer codes constructible on 2D lattices. I was able to show, under a reasonable set of assumptions, that the toric codes, color codes and their generalizations were the only such codes. This research piqued my curiosity in topological quantum computing as well as graph theory.
Additionally, I am interested in probing the divide between classical and quantum computers. Topics in this area that I find interesting are: quantum circuit simulation and classical simulation of quantum systems. Through interest in the latter I have started studying methods for simulating many-body systems such as MPS, PEPS, MERA, etc. The question: What is the computational power of systems found in Nature, I believe, is one of the most fundamental and interesting questions in modern physics.
- Date:
- Speakers:
- Jonas Anderson
- Affiliation:
- University of New Mexico
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Jeff Running
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