Nonbinary Quantum Stabilizer Codes
Alexei Ashikhmin, Emanuel Knill
TL;DR
The paper addresses constructing nonbinary quantum stabilizer codes (p^m-ary) by extending the stabilizer formalism with explicit nonbinary error bases. It develops a $p^m$-ary error basis from generalized Pauli operators and a symplectic-type inner product, then shows how to derive quantum codes from classical selforthogonal codes over $\mathbb{F}_{p^{2m}}$. A concrete construction relates a classical selforthogonal code to a quantum stabilizer code with parameters $[[n,mn-r]]_{p^m}$ and distance tied to $C^{\perp}\setminus C$, leveraging a trace-based inner product to ensure selforthogonality. The approach connects to known good families (Bierbrauer98) and broadens the repertoire of nonbinary quantum codes, enabling systematic design of higher-dimensional quantum error-correcting codes.
Abstract
We define and show how to construct nonbinary quantum stabilizer codes. Our approach is based on nonbinary error bases. It generalizes the relationship between selforthogonal codes over $GF_{4}$ and binary quantum codes to one between selforthogonal codes over $GF_{q^2}$ and $q$-ary quantum codes for any prime power $q$.