Quotient Space Quantum Codes

TL;DR

This paper provides the necessary and sufficient conditions for constructing union stabilizer codes and establishes the quotient space codes to construct quantum codes, and presents new bounds for quantum codes and provides a simple proof of the quantum Singleton bound.

Abstract

Additive codes and some nonadditive codes use the single and multiple invariant subspaces of the stabilizer G, respectively, to construct quantum codes, so the selection of the invariant subspaces is a key problem. In this paper, I provide the necessary and sufficient conditions for this problem and, establish the quotient space codes to construct quantum codes. These new codes unify additive codes and codeword stabilized codes and can transmit classical codewords. Actually, I give an alternative approach to constructing union stabilizer codes, which is different from that of Markus Grassl and Martin Roetteler, and which is easier to deal with degenerate codes. I also present new bounds for quantum codes and provide a simple proof of the quantum Singleton bound. The quotient space approach provides a concise and clear mathematical framework for the study of quantum error-correcting codes.

Theorems & Definitions (10)

• Theorem 1
• proof
• Proposition 1
• proof
• Theorem 2
• proof
• Theorem 3: Hamming type
• Theorem 4: G-V type
• Theorem 5: Singleton
• proof