On the Structure of Additive Quantum Codes and the Existence of Nonadditive Codes

TL;DR

This paper characterizes additive quantum codes, introduces numerous nonadditive codes with high rates and error correction capabilities, and defines strongly nonadditive codes with a specific construction.

Contribution

It provides a characterization of stabilizer codes, demonstrates the existence of infinitely many nonadditive codes with optimal rates, and introduces the concept and construction of strongly nonadditive codes.

Findings

Nonadditive codes can achieve high rates and error correction thresholds.

Existence of infinitely many nonadditive codes with various minimum distances.

Construction of a specific strongly nonadditive ((11,2,3)) code.

Abstract

We first present a useful characterization of additive (stabilizer) quantum error-correcting codes. Then we present several examples of We first present a useful characterization of additive (stabilizer) quantum error--correcting codes. Then we present several examples of nonadditive codes. We show that there exist infinitely many non-trivial nonadditive codes with different minimum distances, and high rates. In fact, we show that nonadditive codes that correct t errors can reach the asymptotic rate R=1-2H(2t/n), where H(x) is the binary entropy function. Finally, we introduce the notion of strongly nonadditive codes (i.e., quantum codes with the following property: the trivial code consisting of the entire Hilbert space is the only additive code that is equivalent to any code containing the given code), and provide a construction for an ((11,2,3)) strongly nonadditive code.