## Description

An \(((n,K,d))_q\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code.

A Galois-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_q\) or \([[n,k,d]]_q\). This notation differentiates between Galois-qudit and modular-qudit \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer codes, although the same notation is usually used for both. Galois-qudit stabilizer codes need not encode an integer number of qudits, with \(K=q^{n-\frac{r}{m}}\), where \(r\) is the number of generators of the stabilizer group, and \(q=p^m\) given prime \(p\) for all Galois qudits. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\).

Galois symplectic representation: The single Galois-qudit Pauli string \(X_{a} Z_{b}\) for \(a,b\in GF(q)\) is converted to the vector \((a|b)\in GF(q)^2\). The multi Galois-qudit version follows naturally.

A pair of Galois-qudit stabilizers on \(n\) Galois qudits with Galois symplectic representation vectors \((a|b)\) and \((a^{\prime}|b^{\prime})\) commute iff their trace symplectic inner product is zero, \begin{align} \text{tr}(a \cdot b^{\prime} - a^{\prime}\cdot b) = \sum_{j=1}^{n} \text{tr}(a_j b^{\prime}_j - a^{\prime}_i b_i) = 0~. \tag*{(1)}\end{align} Galois symplectic representations of stabilizer group elements form a trace-symplectic self-orthogonal linear code over \(GF(q)^{2n}\). The trace-symplectic inner product reduces to the symplectic inner product when the field trace is removed, and a symplectic self-orthogonal set of vectors is automatically trace-symplectic self-orthogonal.

Another correspondence between Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\) yields the one-to-one correspondence between Galois-qudit stabilizer codes and trace-alternating self-orthogonal additive codes over \(GF(q^2)\) [2].

\(GF(q^2)\) representation: An \(n\)-qubit Galois-qudit Pauli stabilizer can be represented as a length-\(n\) vector over \(GF(q^2)\) using the one-to-one correspondence between the \(q^2\) Galois-qudit Pauli matrices and elements of \(GF(q^2)\). Given a basis \((\beta,\beta^q)\) for \(GF(q^2)\) over \(GF(q)\), the vector \((a|b)\in GF(q)^2\) (representing a Galois-qudit Pauli string in the Galois symplectic representation) is in one-to-one correspondence with element \(a \beta + b \beta^q \in GF(q^2)\) [2][3; Thm. 27.3.8].

The sets of \(GF(q^2)\)-represented vectors for all generators yield a trace-alternating self-orthogonal additive code over \(GF(q^2)\).

Galois-qudit stabilizer codes can equivalently [4] (see also [5,6]) be defined using graphs, yielding an analytical form for the codewords [7].

## Protection

## Encoding

## Gates

## Decoding

## Notes

## Parents

- Galois-qudit USt code — A Galois-qudit stabilizer code with stabilizer group \(\mathsf{S}\) can be thought of as a Galois-qudit USt with only the identity coset representative. Conversely, if \(K = q^k\), and if the set of coset representatives of a Galois-qudit USt form a \(q\)-ary linear code, then they can be absorbed into a Galois-qudit stabilizer group that defines the USt.
- Stabilizer code

## Child

## Cousins

- Galois-qudit CWS code — Galois-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code are Galois-qudit stabilizer codes containing a cluster-state codeword.
- Modular-qudit stabilizer code — Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [1]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits. A modular-qudit stabilizer code with composite dimension \(q\) contains a subcode that is isomorphic to a \(p\)-dimensional prime-qudit stabilizer code for every prime factor \(p\) of \(q\), and the distance of the full stabilizer code is bounded by the distance of this subcode [13].
- Additive \(q\)-ary code — Galois-qudit stabilizer codes are the closest quantum analogues of additive codes over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
- Dual additive code — Galois-qudit stabilizer codes are in one-to-one correspondence with trace-symplectic self-orthogonal additive codes of length \(2n\) over \(GF(q)\) via the Galois symplectic representation [1]. They are also in one-to-one correspondence with trace-alternating self-orthogonal additive codes of length \(n\) over \(GF(q^2)\) via the \(GF(q^2)\) representation.
- Graph quantum code — Graph quantum codes for \(G=GF(q)\) are a subset of Galois-qudit stabilizer codes [4]. Any Galois-qubit stabilizer code is equivalent to a graph quantum code for \(G=GF(q)\) via a single-Galois-qudit Clifford circuit [4] (see also [5,6]).
- EA Galois-qudit stabilizer code — EA Galois-qudit stabilizer codes utilize additional ancillary Galois-qudits in a pre-shared entangled state, but reduce to Galois-qudit stabilizer codes when said qudits are interpreted as noiseless physical qudits. Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes [16][17; Thm. 10].
- Subsystem Galois-qudit stabilizer code — Subsystem Galois-qudit stabilizer codes reduce to Galois-qudit stabilizer codes when there are no gauge qudits.

## References

- [1]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [2]
- A. Ketkar et al., “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
- [3]
- M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
- [4]
- D. Schlingemann, “Stabilizer codes can be realized as graph codes”, (2001) arXiv:quant-ph/0111080
- [5]
- M. Van den Nest, J. Dehaene, and B. De Moor, “Graphical description of the action of local Clifford transformations on graph states”, Physical Review A 69, (2004) arXiv:quant-ph/0308151 DOI
- [6]
- M. Grassl, A. Klappenecker, and M. Rotteler, “Graphs, quadratic forms, and quantum codes”, Proceedings IEEE International Symposium on Information Theory, arXiv:quant-ph/0703112 DOI
- [7]
- D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI
- [8]
- M. Grassl and M. Rotteler, “Quantum MDS codes over small fields”, 2015 IEEE International Symposium on Information Theory (ISIT) (2015) arXiv:1502.05267 DOI
- [9]
- M. GRASSL, M. RÖTTELER, and T. BETH, “EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES”, International Journal of Foundations of Computer Science 14, 757 (2003) arXiv:quant-ph/0211014 DOI
- [10]
- P. J. Nadkarni and S. S. Garani, “Quantum error correction architecture for qudit stabilizer codes”, Physical Review A 103, (2021) DOI
- [11]
- M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
- [12]
- M. F. Ezerman et al., “Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes”, (2024) arXiv:2405.15057
- [13]
- Markus Grassl, private communication, 2024
- [14]
- D. Gross, “Hudson’s theorem for finite-dimensional quantum systems”, Journal of Mathematical Physics 47, (2006) arXiv:quant-ph/0602001 DOI
- [15]
- “Nonbinary Stabilizer Codes”, Mathematics of Quantum Computation and Quantum Technology 305 (2007) DOI
- [16]
- L. Riguang and M. Zhi, “Non-binary Entanglement-assisted Stabilizer Quantum Codes”, (2011) arXiv:1105.5872
- [17]
- M. Grassl, F. Huber, and A. Winter, “Entropic Proofs of Singleton Bounds for Quantum Error-Correcting Codes”, IEEE Transactions on Information Theory 68, 3942 (2022) arXiv:2010.07902 DOI

## Page edit log

- Markus Grassl (2024-07-11) — most recent
- Victor V. Albert (2022-07-22)
- Victor V. Albert (2022-04-13)
- Leonid Pryadko (2022-04-13)
- Victor V. Albert (2022-01-12)
- Qingfeng (Kee) Wang (2022-01-07)

## Cite as:

“Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_stabilizer