Algebraic Methods for Quantum Codes on Lattices

Abstract

This is a note from a series of lectures at Encuentro Colombiano de Computacion Cuantica, Universidad de los Andes, Bogota, Colombia, 2015. The purpose is to introduce additive quantum error correcting codes, with emphasis on the use of binary representation of Pauli matrices and modules over a translation group algebra. The topics include symplectic vector spaces, Clifford group, cleaning lemma, an error correcting criterion, entanglement spectrum, implications of the locality of stabilizer group generators, and the classification of translation-invariant one-dimensional additive codes and two-dimensional CSS codes with large code distances. In particular, we describe an algorithm to find a Clifford quantum circuit (CNOTs) to transform any two-dimensional translation-invariant CSS code on qudits of a prime dimension with code distance being the linear system size, into a tensor product of finitely many copies of the qudit toric code and a product state. Thus, the number of embedded toric codes is the complete invariant of these CSS codes under local Clifford circuits.

54 pages; (v2) ref. added

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• Bipartite entanglement and entropic boundary law in lattice spin systems
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• Fracton fusion and statistics
• Sorting topological stabilizer models in three dimensions
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• Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices
• Entanglement Spectra of Stabilizer Codes: A Window into Gapped Quantum Phases of Matter
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• Spatially-Coupled QLDPC Codes
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• A graph-based formalism for surface codes and twists
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• Decomposition of Pauli groups via weak central products
• A stabilizer code model with non-invertible symmetries: Strange fractons, confinement, and non-commutative and non-Abelian fusion rules
• Fracton models from product codes