Modular-qudit code


Also called a \(\mathbb{Z}_q\)-qudit code. Encodes \(K\)-dimensional Hilbert space into a \(q^n\)-dimensional (\(n\)-qudit) Hilbert space, with canonical qudit states \(|k\rangle\) labeled by elements \(k\) of the group \(\mathbb{Z}_q\) of integers modulo \(q\). Usually denoted as \(((n,K))_q\) or \(((n,K,d))_q\), whenever the code's distance \(d\) is defined, and with \(q=p\) when the dimension is prime.


A convenient and often considered error set is the modular-qudit analogue 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 ~, \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 [1]. 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.


For few-qudit codes (\(n\) is small), decoding can be based on a lookup table. For infinite code families, the size of such a table scales exponentially with \(n\), so approximate decoding algorithms scaling polynomially with \(n\) have to be used. The decoder determining the most likely error given a noise channel is called the maximum-likelihood decoder.




  • Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [2]; see Sec. 5.3 of Ref. [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.
  • Group-based quantum code — Group quantum codes whose physical spaces are constructed using modular-integer groups \(\mathbb{Z}_q\) are modular-qudit codes.

Zoo code information

Internal code ID: qudits_into_qudits

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Zoo Code ID: qudits_into_qudits

Cite as:
“Modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_qudits_into_qudits, title={Modular-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”. quant-ph/9608048
Alexei Ashikhmin and Emanuel Knill, “Nonbinary Quantum Stabilizer Codes”. quant-ph/0005008
Annika Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”. quant-ph/0501074

Cite as:

“Modular-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.