Description
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 Galois field \(GF(q)\) and with \(q\) being a power of a prime \(p\).
A Galois field can be thought of as a vector space whose basis vectors are the \(m\) roots of some polynomial and whose coefficients (i.e., field) are \(p\)th roots of unity.
Codes can be denoted as \(((n,K))_q\) or \(((n,K,d))_q\), whenever the code''s distance \(d\) is defined. This notation differentiates between Galois-qudit and \(((n,K,d))_{\mathbb{Z}_q}\) modular-qudit codes, although the same notation is usually used for both.'
There exists an analogue of the Wigner function for Galois qudits [1,2].
Protection
Galois-qudit Pauli-string error basis
A convenient and often considered error set is the Galois-qudit analogue of the Pauli string set for qubit codes.
Galois-qudit Pauli strings: For a single Galois qudit, this set consists of products of \(X\)-type and \(Z\)-type operators labeled by elements \(\beta \in GF(q)\), which act on computational basis states \(|\gamma\rangle\) for \(\gamma\in GF(q)\) as \begin{align} X_{\beta}\left|\gamma\right\rangle =\left|\gamma+\beta\right\rangle \,\,\text{ and }\,\,Z_{\beta}\left|\gamma\right\rangle =e^{i\frac{2\pi}{p}\text{tr}(\beta\gamma)}\left|\gamma\right\rangle~, \tag*{(1)}\end{align} where \(\text{tr}\) is the field trace. For multiple Galois qudits, error set elements are tensor products of elements of the single-qudit error set. Tensor products of \(X\) (\(Z\)) Galois-qudit Paulis acting on different qudits are called \(X\)-type (\(Z\)-type) Galois-qudit Pauli strings. Combining the \(X\)-type and \(Z\)-type strings with a \(p\)th root of unity forms a group called the Galois-qudit Pauli group (on \(n\) Galois qudits.
The Galois-qudit 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 Galois qudit Pauli string that implements a nontrivial logical operation in the code.
Gates
The normalizer of the Galois-qudit Pauli group is the Galois-qudit Clifford group [3].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 (ML) decoder.Cousins
- Modular-qudit code— A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits; see [5,7–9][6; 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.
- EA Galois-qudit code— EA Galois-qudit codes utilize additional ancillary Galois qudits in a pre-shared entangled state, but reduce to ordinary Galois-qudit codes when said qudits are interpreted as noiseless physical qudits.
- Subsystem Galois-qudit code— Subsystem Galois-qudit codes reduce to (subspace) Galois-qudit codes when there is no gauge subsystem.
Member of code lists
Primary Hierarchy
References
- [1]
- K. S. Gibbons, M. J. Hoffman, and W. K. Wootters, “Discrete phase space based on finite fields”, Physical Review A 70, (2004) arXiv:quant-ph/0401155 DOI
- [2]
- J. P. Paz, A. J. Roncaglia, and M. Saraceno, “Qubits in phase space: Wigner-function approach to quantum-error correction and the mean-king problem”, Physical Review A 72, (2005) arXiv:quant-ph/0410117 DOI
- [3]
- M. GRASSL, M. RÖTTELER, and T. BETH, “EFFICIENT QUANTUM CIRCUITS FOR NON-QUBIT QUANTUM ERROR-CORRECTING CODES”, International Journal of Foundations of Computer Science 14, 757 (2003) arXiv:quant-ph/0211014 DOI
- [4]
- Eric Sabo. CodingTheory. https://github.com/esabo/CodingTheory
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- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [6]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [7]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [8]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [9]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
- [10]
- R. Zucchini, “Calibrated hypergraph states: II calibrated hypergraph state construction and applications”, (2025) arXiv:2501.18968
- [11]
- Y. Zhang, “Quantum Fourier Transform Over Galois Rings”, (2009) arXiv:0904.2560
Page edit log
- Victor V. Albert (2022-05-07) — most recent
- Victor V. Albert (2021-12-03)
Cite as:
“Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_into_galois