# Modular-qudit code

## Description

## Protection

A convenient and often considered error set is the modular-qudit analogue [1][2] of the Pauli string basis for qubit codes. For a single qudit, this set consists of products of powers of the qudit Pauli matrices \(X\) and \(Z\), which act on computational basis states \(|k\rangle\) for \(k\in\mathbb{Z}_q\) as \begin{align} X\left|k\right\rangle =\left|k+1\right\rangle \,\,\text{ and }\,\,Z\left|k\right\rangle =e^{i\frac{2\pi}{q}k}\left|k\right\rangle ~, \tag*{(1)}\end{align} with addition performed modulo \(q\). For multiple qudits, error set elements are tensor products of elements of the single-qudit error set.

The Pauli error set is a unitary basis for linear operators on the multi-qudit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a nice error basis [3][4][5]. The distance associated with this set is often the minimum weight of a qudit Pauli string that implements a nontrivial logical operation in the code.

## Decoding

## Notes

## Parents

- Block quantum code
- Finite-dimensional quantum error-correcting code
- Group-based quantum code — Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.
- Category-based quantum code — Category quantum codes whose physical spaces are constructed using \(\mathbb{Z}_q\) as the category are modular-qudit codes.

## Children

- Qubit code — Modular-qudit quantum codes for \(q=2\) correspond to qubit codes.
- Modular-qudit stabilizer code

## Cousins

- Entanglement-assisted (EA) QECC — Pure modular-qudit codes can be used to make EA-QECCs with the same distance and dimension; see Thm. 10 of Ref. [9].
- Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [10]; see Sec. 5.3 of Ref. [11]. The two coincide when \(q\) is prime, and reduce to qubits when \(q=2\). However, Pauli matrices for the two types of qudits are defined differently.

## References

- [1]
- “Full length article”, Chaos, Solitons & Fractals 10, 1749 (1999) arXiv:quant-ph/9802007 DOI
- [2]
- R. Sarkar and T. J. Yoder, “The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems”, (2023) arXiv:2302.07966
- [3]
- E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608048
- [4]
- E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
- [5]
- A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes I: Nice Error Bases”, (2001) arXiv:quant-ph/0010082
- [6]
- E. Hostens, J. Dehaene, and B. De Moor, “Stabilizer states and Clifford operations for systems of arbitrary dimensions and modular arithmetic”, Physical Review A 71, (2005) arXiv:quant-ph/0408190 DOI
- [7]
- A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004) arXiv:quant-ph/0310137 DOI
- [8]
- L. Schatzki et al., “A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement”, (2022) arXiv:2209.07607
- [9]
- 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
- [10]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [11]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074

## Page edit log

- Victor V. Albert (2022-05-07) — most recent
- Victor V. Albert (2021-10-29)

## Cite as:

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