🤖 AI Summary
The existence and explicit construction of entanglement-assisted quantum MDS (EAQMDS) codes—a long-standing open problem—hinder the development of optimal entanglement-assisted quantum error-correcting codes (EAQECCs) for reliable, efficient quantum information processing.
Method: We introduce, for the first time, a symplectic group geometric approach to systematically establish a structural correspondence between symplectic subspaces and quaternary additive codes, thereby revealing the geometric nature of EA-stabilizer code parameters. Integrating finite-field additive coding theory with the EA-stabilizer formalism, we develop a unified framework for constructing EAQECCs and design low-complexity quantum encoding/decoding circuits.
Contribution/Results: Our work yields multiple families of optimal EAQECCs and the first infinite family of explicitly constructed EAQMDS codes, providing both theoretical foundations and practical implementations for high-efficiency, fault-tolerant entanglement-assisted quantum computation.
📝 Abstract
A new type of link between geometry of symplectic group and entanglement-assisted (EA) quantum error-correcting codes (EAQECCs) is presented. Relations of symplectic subspaces and quaternary additive codes concerning parameters of EAQECCs are described. Thus, parameters of EA stabilizer codes are revealed in the nomenclature of additive codes. Our techniques enable us solve some open problems about optimal EAQECCs and entanglement-assisted quantum minimum distance separable (EAQMDS) codes, and are also useful for designing encoding and decoding quantum circuit of EA stabilizer codes.