True Galois-qudit stabilizer code

True Galois-qudit stabilizer code[1–3]

Alternative names: Linear stabilizer code.

Description

A \([[n,k,d]]_q\) stabilizer code whose stabilizer's Galois symplectic representation forms a linear subspace. In other words, the set of \(q\)-ary vectors representing the stabilizer group is closed under both addition and multiplication by elements of \(GF(q)\). In contrast, Galois-qudit stabilizer codes admit sets of vectors that are closed under addition only.

The number of generators \(r\) for a true stabilizer code is a multiple of \(m\) (recall that \(q=p^m\) for Galois qudits). As a result, the number \(k=n-r/m\) of logical qudits is an integer.

Each code can be represented by a stabilizer generator matrix \(H=(A|B)\), where each row \((a|b)\) is the Galois symplectic representation of a stabilizer generator.

Protection

Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits.

Notes

See Ref. [4,5] for introductions to various stabilizer code constructions.

Cousin

Linear \(q\)-ary code— A true Galois-qudit stabilizer code is the closest quantum analogue of a linear code over \(GF(q)\) because the \(q\)-ary vectors corresponding to the Galois symplectic representation of the stabilizers form a linear subspace.

Member of code lists

Primary Hierarchy

Quantum Domain

Galois-qudit Kingdom

Galois-qudit codeQECC Quantum

Galois-qudit USt code

Galois-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

Parents

Galois-qudit stabilizer code

True Galois-qudit stabilizer code

Children

Qubit stabilizer codeBB Homological Fermion-into-qubit Quantum Reed-Muller Color Twist-defect color CDSC Twist-defect surface Kitaev surface

True Galois-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes.

\([[5,1,3]]_q\) Galois-qudit code

\([[6,2,3]]_{q}\) code

The code is a non-CSS stabilizer code in general [6].

\([[7,3,3]]_{q}\) code

Galois-qudit BCH code

Galois-qudit BCH codes can be constructed via the CSS construction or the Hermitian construction.

Galois-qudit CSS codeQuantum Reed-Muller Color Homological Quantum QR code BB Kitaev surface

Galois-qudit CSS codes are true stabilizer codes [3].

Quantum duadic codeQuantum QR code

Quantum duadic codes can be constructed via the CSS construction or the Hermitian construction.

Galois-qudit quantum RM codeQuantum Reed-Muller

Galois-qudit RM codes can be constructed via the CSS construction or the Hermitian construction.

Galois-qudit GRS code

Galois-qudit GRS codes can be constructed via the CSS construction or the Hermitian construction.

Quantum Gabidulin code

Hermitian Galois-qudit code

Hermitian codes are true stabilizer codes because they are based on Hermitian self-orthogonal linear (as opposed to additive) codes over \(GF(q^2)\).

References

[1]: A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
[2]: A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, “Nonbinary stabilizer codes over finite fields”, (2005) arXiv:quant-ph/0508070
[3]: D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
[4]: A. Klappenecker, “Algebraic quantum coding theory”, Quantum Error Correction 307 (2013) DOI
[5]: M. F. Ezerman, "Quantum Error-Control Codes." Concise Encyclopedia of Coding Theory (Chapman and Hall/CRC, 2021) DOI
[6]: Z. Wang, S. Yu, H. Fan, and C. H. Oh, “Quantum error-correcting codes over mixed alphabets”, Physical Review A 88, (2013) arXiv:1205.4253 DOI

Page edit log

Victor V. Albert (2022-07-22) — most recent
Daniel Gottesman (2022-02-17)
Victor V. Albert (2022-02-17)

Cite as:

“True Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_true_stabilizer

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/stabilizer/galois_true_stabilizer.yml.