\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code

\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code[1,2]

Description

Modular-qudit stabilizer code that generalizes the five-qubit perfect code using properties of the multiplicative group \(\mathbb{Z}_q\) [1]; see also [2; Thm. 13]. It has four stabilizer generators consisting of \(X Z Z^\dagger X^\dagger I\) and its cyclic permutations.

The components of the encoding isometry in the computational basis (with \(a\) being the logical qubit index) are [3; Sec. VI.B] \begin{align} T_{aklmnp}=\delta_{a,k+l+m+n+p}^{\mathbb{Z}_{q}}\frac{1}{q^{2}}\omega^{kl+lm+mn+np+pk}~, \tag*{(1)}\end{align} where \(\omega\) is a primitive \(q\)th root of unity, and where \(\delta^{\mathbb{Z}_{q}}\) is the \(\mathbb{Z}_q\) Kronecker-delta function.

Protection

Protects against a single error on any one qudit. Detects two-qudit errors.

Encoding

Generalized CNOT, Toffoli, and quantum Fourier transform gates.Encoder for prime-dimensional qudits [4].

Gates

Magic-state distillation for the \(q=3\) case [5].

Decoding

Decoder for prime-dimensional qudits [4].

Cousins

Five-rotor code— The five-rotor code is a rotor analogue of the five-qudit code.
\([[10,1,4]]_{G}\) tenfold code— The \([[10,1,4]]_{G}\) Abelian group code for \(G=\mathbb{Z}_q\) is defined using a graph that is closely related to the \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code [6].
\([[5,1,3]]_{\mathbb{R}}\) Braunstein five-mode code— The Braunstein five-mode code is a bosonic analogue of the five-qudit code.

Member of code lists

Primary Hierarchy

Quantum Domain

Modular-qudit Kingdom

Modular-qudit codeQECC Quantum

Modular-qudit USt code

Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

Parents

Modular-qudit stabilizer code

Perfect-tensor codeQECC Quantum

The \([[5,1,3]]_{\mathbb{Z}_q}\) code is a perfect-tensor code because it stems from the \([[6,0,4]]_{\mathbb{Z}_q}\) AME state [2; Thm. 13].

Graph quantum codeStabilizer Hamiltonian-based QECC Quantum

The \([[5,1,3]]_{\mathbb{Z}_q}\) code is equivalent via a single-modular-qudit Clifford circuit to a graph quantum code for the group \(G=Z_q\) [6].

Cyclic quantum codeQECC Quantum

Small-distance block quantum codeQECC Quantum

\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code

Children

Five-qubit perfect code

The \([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code for \(q=2\) reduces to the five-qubit perfect code.

References

[1]: H. F. Chau, “Five quantum register error correction code for higher spin systems”, Physical Review A 56, R1 (1997) arXiv:quant-ph/9702033 DOI
[2]: E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
[3]: P. Faist, S. Nezami, V. V. Albert, G. Salton, F. Pastawski, P. Hayden, and J. Preskill, “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
[4]: J. Keppens, Q. Eggerickx, V. Levajac, G. Simion, and B. Sorée, “Qudit vs. Qubit: Simulated performance of error correction codes in higher dimensions”, (2025) arXiv:2502.05992
[5]: H. Anwar, E. T. Campbell, and D. E. Browne, “Qutrit magic state distillation”, New Journal of Physics 14, 063006 (2012) arXiv:1202.2326 DOI
[6]: D. Schlingemann and R. F. Werner, “Quantum error-correcting codes associated with graphs”, Physical Review A 65, (2001) arXiv:quant-ph/0012111 DOI

Page edit log

Sarah Meng Li (2022-02-21) — most recent
Victor V. Albert (2022-02-21)
Victor V. Albert (2023-01-14)

Cite as:

“\([[5,1,3]]_{\mathbb{Z}_q}\) modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_5_1_3

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/small/qudit_5_1_3.yml.