# Modular-qudit stabilizer code[1]

## Description

An \(((n,K,d))_q\) modular-qudit code whose logical subspace is the joint eigenspace of commuting 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 qudit Pauli string that implements a nontrivial logical operation in the code.

A modular-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_{\mathbb{Z}_q}\) or \([[n,k,d]]_{\mathbb{Z}_q}\). For composite \(q\), such codes need not encode an integer number of qudits, with \(K=q^n/|\mathsf{S}|\) [2]. This is because \(|{\mathsf{S}}|\) need not be a power of \(q\), as group generators may have different orders. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\). Prime-qudit stabilizer codes, where \(q=p\) for some prime \(p\), do not suffer from this issue and encode \(n-k\) logical qudits, with \(K=p^{n-k}\).

Each code can be represented by a check matrix (a.k.a. stabilizer generator matrix) \(H=(A|B)\), where each row \((a|b)\) is the \(q\)-ary symplectic representation of a stabilizer generator. The check matrix can be brought into standard form via Gaussian elimination [2].

One can switch between stabilizer codes by appending another abelian subgroup of the Pauli group to the stabilizer group and taking the center of the resulting larger group.

Code switching: Code switching is a map between stabilizer codes that is done using a stabilizer group \(\mathsf{F}\) of the \(n\)-qudit Pauli group, \begin{align} \mathsf{S}\to\mathsf{N}_{\left\langle \mathsf{S},\mathsf{F}\right\rangle }\left(\mathsf{F}\right)~, \tag*{(1)}\end{align} where \(\mathsf{Z}\) denotes taking the center of a group. Code switching may not preserve the logical information and instead implement logical measurements; conditions on \(\mathsf{S}\) and \(\mathsf{F}\) such that qubit stabilizer code switching preserves logical information are derived in [3; Prop. II.1]. In the context of abelian topological stabilizer codes, code switching implements anyon condensation of any anyons represented by operators in the group \(\mathsf{F}\).

## Protection

## Decoding

## Notes

## Parents

- Modular-qudit code
- Stabilizer code
- Quantum Lego code — Modular-qudit stabilizer codes are quantum Lego codes built out of atomic blocks such as the 2-qudit repetition code, single-qudit trivial stabilizer codes, and tensor-products of the \(|0\rangle\) state.

## Children

- Qubit stabilizer code — Modular-qudit stabilizer codes for \(q=2\) correspond to qubit stabilizer codes. Modular-qudit stabilizer codes for prime-dimensional qudits \(q=p\) inherit most of the features of qubit stabilizer codes, including encoding an integer number of qudits and a Pauli group with a unique number of generators. Conversely, qubit codes can be extended to modular-qudit codes by decorating appropriate generators with powers. For example, \([[4,2,2]]\) qubit code generators can be adjusted to \(ZZZZ\) and \(XX^{-1} XX^{-1}\). A systematic procedure extending a qubit code to prime-qudit codes involves putting its generator matrix into local-dimension-invariant (LDI) form [6]. Various bounds exist on the distance of the resulting codes [7,8].
- Frobenius code
- Modular-qudit CSS code
- \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code
- Subsystem modular-qudit stabilizer code — Subsystem modular-qudit stabilizer codes reduce to modular-qudit stabilizer codes when there are no gauge qudits.
- Abelian TQD stabilizer code — All Abelian TQD codes can be realized as modular-qudit stabilizer codes by starting with an abelian quantum double model along with a family of Abelian TQDs that generalize the double semion anyon theory and condensing certain bosonic anyons [9].

## Cousins

- \(q\)-ary code over \(\mathbb{Z}_q\) — Modular-qudit stabilizer codes are the closest quantum analogues of additive codes over \(\mathbb{Z}_q\) because addition in the ring corresponds to multiplication of stabilizers in the quantum case.
- Analog stabilizer code — Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [10].
- Translationally invariant stabilizer code — Modular-qudit stabilizer codes can be thought of as translationally-invariant stabilizer codes for dimension \(D = 0\), with the lattice consisting of a single site.
- Galois-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 \(k=nm-r\) [11]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits.

## References

- [1]
- D. Gottesman, “Stabilizer Codes and Quantum Error Correction”, (1997) arXiv:quant-ph/9705052
- [2]
- V. Gheorghiu, “Standard form of qudit stabilizer groups”, Physics Letters A 378, 505 (2014) arXiv:1101.1519 DOI
- [3]
- D. Aasen et al., “Measurement Quantum Cellular Automata and Anomalies in Floquet Codes”, (2023) arXiv:2304.01277
- [4]
- E. Sabo, A. B. Aloshious, and K. R. Brown, “Trellis Decoding For Qudit Stabilizer Codes And Its Application To Qubit Topological Codes”, (2022) arXiv:2106.08251
- [5]
- T. Singal et al., “Counting stabiliser codes for arbitrary dimension”, Quantum 7, 1048 (2023) arXiv:2209.01449 DOI
- [6]
- L. G. Gunderman, “Local-dimension-invariant qudit stabilizer codes”, Physical Review A 101, (2020) arXiv:1910.08122 DOI
- [7]
- A. J. Moorthy and L. G. Gunderman, “Local-dimension-invariant Calderbank-Shor-Steane Codes with an Improved Distance Promise”, (2021) arXiv:2110.11510
- [8]
- L. G. Gunderman, “Degenerate local-dimension-invariant stabilizer codes and an alternative bound for the distance preservation condition”, Physical Review A 105, (2022) arXiv:2110.15274 DOI
- [9]
- T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, (2022) arXiv:2211.03798
- [10]
- L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, (2023) arXiv:2303.17000
- [11]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI

## Page edit log

- Victor V. Albert (2022-05-19) — most recent
- Victor V. Albert (2022-02-16)
- Leonid Pryadko (2022-02-16)
- Qingfeng (Kee) Wang (2022-01-07)
- Victor V. Albert (2021-11-02)

## Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits/qudit_stabilizer.yml.