## Description

## Protection

### Modular-qudit Pauli-string error basis

A convenient and often considered error set is the modular-qudit analogue [1,2] of the Pauli string basis for qubit codes.

Modular-qudit Pauli strings: 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. Modular-qudit Pauli matrices [3,4] are also known as Weyl operators [5], Sylvester-t'Hooft generators [6,7], or clock and shift matrices [8]; they are special cases of Manin's quantum plane [9]

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. 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.

## Gates

Qudit Clifford hierarchy: The modular-qudit Clifford hierarchy [19–22] is a tower of gate sets which includes modular-qudit Pauli and modular-qudit Clifford gates at its first two levels, and non-Clifford qudit gates at higher levels. The \(k\)th level is defined recursively by \begin{align} C_k = \{ U | U P U^{\dagger} \in C_{k-1} \}~, \tag*{(2)}\end{align} where \(P\) is any modular-qudit Pauli matrix, and \(C_1\) is the modular-qudit Pauli group.

## 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 the group \(\mathbb{Z}_q\) as the category are modular-qudit codes.

## Children

- Qubit code — Modular-qudit quantum codes for \(q=2\) correspond to qubit codes.
- \(((3,6,2))_{\mathbb{Z}_6}\) code
- Modular-qudit USt 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. [25].
- Bosonic \(q\)-ary expansion — The bosonic \(q\)-ary expansion allows one to map between prime-dimensional qudit states and a Fock subspace of a single mode.
- Subsystem modular-qudit code — Subsystem modular-qudit codes reduce to (subspace) modular-qudit codes when there is no gauge subsystem.
- Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [26]; see Sec. 5.3 of Ref. [27]. 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

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- [2]
- R. Sarkar and T. J. Yoder, “The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems”, Quantum 8, 1307 (2024) arXiv:2302.07966 DOI
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- J. Schwinger, Quantum Kinematics and Dynamics (CRC Press, 2018) DOI
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- Weyl, Hermann. The theory of groups and quantum mechanics. Courier Corporation, 1950.
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- Sylvester, James Joseph. The Collected Mathematical Papers of James Joseph Sylvester... Vol. 3. University Press, 1909.
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- Yu. I. Manin, “Some remarks on Koszul algebras and quantum groups”, Annales de l’institut Fourier 37, 191 (1987) DOI
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- M. Howard and J. Vala, “Qudit versions of the qubitπ/8gate”, Physical Review A 86, (2012) arXiv:1206.1598 DOI
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- S. X. Cui, D. Gottesman, and A. Krishna, “Diagonal gates in the Clifford hierarchy”, Physical Review A 95, (2017) arXiv:1608.06596 DOI
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- 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
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- L. Schatzki et al., “A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement”, (2022) arXiv:2209.07607
- [25]
- 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
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- 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