Modular-qudit code

Description

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.

Protection

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.

Decoding

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.

Notes

See Refs. [2][3] for descriptions of the qudit Clifford group.Weight distribution of a code depends on the average entanglement of codewords [4][5].

Parent

Child

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. [6].
  • Galois-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [7]; see Sec. 5.3 of Ref. [8]. 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.

References

[1]
E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”. quant-ph/9608048
[2]
“Full length article”, Chaos, Solitons & Fractals 10, 1749 (1999). DOI; quant-ph/9802007
[3]
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). DOI; quant-ph/0408190
[4]
A. J. Scott, “Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions”, Physical Review A 69, (2004). DOI; quant-ph/0310137
[5]
Louis Schatzki et al., “A Hierarchy of Multipartite Correlations Based on Concentratable Entanglement”. 2209.07607
[6]
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). DOI; 2010.07902
[7]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001). DOI
[8]
Annika Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”. quant-ph/0501074
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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. https://errorcorrectionzoo.org/c/qudits_into_qudits
BibTeX:
@incollection{eczoo_qudits_into_qudits, title={Modular-qudit code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/qudits_into_qudits} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits/qudits_into_qudits.yml.