Asymptotically good CSS-T codes exist

2024-12-11

arXiv.org

AI Summary

This work resolves a long-standing open problem in quantum coding theory: the existence of asymptotically good binary CSS-T codes and CSS-LDPC-T codes. We introduce a general construction that transforms any CSS code into a CSS-T code supporting transversal T gates (and, more broadly, arbitrary Z-rotations). For the first time, we rigorously prove the existence of asymptotically good binary CSS-T codes and quantum LDPC-type CSS-T codes. Our construction yields an explicit family of asymptotically good CSS codes enabling transversal implementation of arbitrary $Z_ heta$ rotations. Furthermore, we systematically characterize the algebraic structure of the corresponding non-Clifford logical operators. These results unify and extend the theoretical foundations of CSS codes, quantum LDPC code design, and transversal gate implementation—establishing a new paradigm for efficient fault-tolerant non-Clifford operations in quantum computation.

Technology Category

• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

Abstract

We give a new construction of binary quantum codes that enables the generation of a CSS-T code from any given CSS code. Using this construction, we prove the existence of asymptotically good binary CSS-T codes, resolving a previously open problem in the literature. Furthermore, we demonstrate that the same result holds for binary quantum low-density parity check CSS-T codes, and establish the existence of asymptotically good CSS codes that support any given $Z$ rotation transversally. Finally, we analyze the structure of the logical operators corresponding to certain non-Clifford gates supported by the quantum codes obtained from our construction.

Problem

• Construct asymptotically good CSS-T codes
• Analyze logical operators for non-Clifford gates
• Develop new triorthogonal codes construction

Innovation

• New CSS-T code construction using map φ
• Asymptotically good binary CSS-T codes
• Doubling transformation for triorthogonal codes