## 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 modular-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. Tensor products of \(X\) (\(Z\)) modular-qudit Paulis acting on different qudits are called \(X\)-type (\(Z\)-type) modular-qudit Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with a primitive \(q\)th root of unity forms a group called the modular-qudit Pauli group (on \(n\) modular qudits.

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
- 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.
- Modular-qudit DA code
- \(((3,6,2))_{\mathbb{Z}_6}\) Euler code
- Modular-qudit USt code

## Cousins

- 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; see [25,27][26; Sec. 5.3]. 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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## 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