2013/09/26 by Daniel Gottesman, Gottesman, Daniel, Lucy Liuxuan Zhang +1
Computer Science · Physics and Astronomy · #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum Mechanics and Applications #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1309.7062
openalex publication_date 2013/09/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a differential geometric framework for describing families of quantum error-correcting codes and for understanding quantum fault tolerance. This work unifies the notion of topological fault tolerance with fault tolerance in other kinds of quantum error-correcting codes. In particular, we use fibre bundles with a natural flat projective connection to study the transformation of codewords under unitary fault-tolerant evolutions. We show that the fault-tolerant logical operations are given by the monodromy group for either of two bundles, both of which have flat projective connections. As concrete realizations of the general framework, we construct the bundles explicitly for two examples of fault-tolerant families of operations, the qudit transversal gates and the string operators in the toric code.