Nonbinary Codeword Stabilized Quantum Codes

TL;DR

This paper extends the codeword stabilized quantum codes framework to nonbinary systems, enabling the search for more efficient quantum error-correcting codes beyond binary states by analyzing their structural properties.

Contribution

It generalizes the CWS formalism to nonbinary dimensions and explores structural differences, opening pathways for constructing superior quantum codes.

Findings

Nonbinary CWS codes have similar additivity properties to binary codes.

Structural properties of nonbinary codes differ significantly from binary codes.

Potential for constructing codes with higher encoding dimensions than existing stabilizer codes.

Abstract

The codeword stabilized (CWS) quantum codes formalism presents a unifying approach to both additive and nonadditive quantum error-correcting codes (arXiv:0708.1021 [quant-ph]), but only for binary states. Here we generalize the CWS framework to the nonbinary case (of both prime and nonprime dimension) and map the search for nonbinary quantum codes to a corresponding search problem for classical nonbinary codes with specific error patterns. We show that while the additivity properties of nonbinary CWS codes are similar to the binary case, the structural properties of the nonbinary codes differ substantially from the binary case, even for prime dimensions. In particular, we identify specific structure patterns of stabilizer groups, based on which efficient constructions might be possible for codes that encode more dimensions than any stabilizer codes of the same length and distance; similar methods cannot be applied in the binary case. Understanding of these structural properties can help prune the search space and facilitate the identification of good nonbinary CWS codes.