Quantum error codes extended for better fault tolerance in computing
Quantum Matrix-Product Codes: CSS-T Characterization and Maximality
Information Theory
Summary
Quantum computers need ways to fix errors that happen during calculations to work properly. The authors study a special class of quantum codes called CSS-T codes that help prevent error spread and allow certain important operations to be done safely. They show how to build bigger and more powerful CSS-T codes by combining smaller ones in a structured way. This helps in creating longer codes that are better suited for protecting quantum information. Their work also gives detailed methods for certain types of codes called cyclic codes to understand and build these error-correcting codes more easily.
quantum error-correcting codesfault-tolerant quantum computationCSS-T codestransversal T-gateclassical binary linear codesSchur squarematrix-product codeu plus v constructioncyclic codescyclotomic sets
Authors
Delio Jaramillo-Velez, Alessandro Neri, Adway Patra, Diego Ruano, Flavio Salizzoni
Abstract
CSS-T codes are quantum error-correcting codes that play an important role in fault-tolerant quantum computation, as they help mitigate the proliferation of errors. They admit a transversal T-gate and are defined from a pair of nested classical binary linear codes satisfying a specific algebraic condition expressed in terms of their Schur square. We extend this algebraic characterization to the propagation rule known as the $(u \mid u+v)$-construction, that is, the matrix-product code constructed from two constituent codes. Moreover, for cyclic constituent codes, we provide an explicit characterization in terms of the defining cyclotomic sets, which extends existing results for cyclic codes. This framework allows the construction of new and longer CSS-T codes.