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.'
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.
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.
Bounds on code parameters
Bounds on code performance include the quantum Singleton bound, quantum Hamming bound, quantum GV bound, various quantum linear programming (LP) bounds [1,2] (see the book [3]), and other bounds [4]. A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code).
Quantum GV bound: The quantum GV bound [5] (see also Refs. [6–9]) for Galois qudits states that a pure \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if \begin{align} \frac{q^{n-k+2}-1}{q^{2}-1}>\sum_{j=1}^{d-1}(q^{2}-1)^{j-1}\binom{n}{j}~. \tag*{(2)}\end{align} The quantum GV bound gives rise to the asymptotic quantum GV bound (i.e., quantum GV bound in the \(n\to\infty\) limit), expressed in terms of the maximum achievable rate \(R\) and relative distance \(\delta\), \begin{align} R\geq 1-\delta\log_q(q+1) - h_{q}(\delta)~, \tag*{(3)}\end{align} where \(h_q\) is the \(q\)-ary entropy function.
Gates
Decoding
Notes
Parents
- Block quantum code
- Finite-dimensional quantum error-correcting code
- Group-based quantum code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [8]; see Sec. 5.3 of Ref. [12]. Interpreted this way, Galois-qudit codes are group quantum codes whose physical spaces are constructed using Galois fields \(GF(q)\) as groups.
- Category-based quantum code — Category quantum codes whose physical spaces are constructed using the group \(GF(q)\) as the category are Galois-qudit codes.
Children
- Qubit code — Galois-qudit quantum codes for \(q=2\) correspond to qubit codes.
- Galois-qudit USt code
Cousins
- Modular-qudit code — A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [8]; see Sec. 5.3 of Ref. [12]. 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.
References
- [1]
- E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
- [2]
- A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
- [3]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [4]
- Keqin Feng, San Ling, and Chaoping Xing, “Asymptotic bounds on quantum codes from algebraic geometry codes”, IEEE Transactions on Information Theory 52, 986 (2006) DOI
- [5]
- K. Feng and Z. Ma, “A Finite Gilbert–Varshamov Bound for Pure Stabilizer Quantum Codes”, IEEE Transactions on Information Theory 50, 3323 (2004) DOI
- [6]
- A. Ekert and C. Macchiavello, “Error Correction in Quantum Communication”, (1996) arXiv:quant-ph/9602022
- [7]
- A. Ashikhmin et al., “Quantum Error Detection II: Bounds”, (1999) arXiv:quant-ph/9906131
- [8]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [9]
- Y. Ma, “The asymptotic probability distribution of the relative distance of additive quantum codes”, Journal of Mathematical Analysis and Applications 340, 550 (2008) DOI
- [10]
- 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
- [11]
- 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
- [12]
- 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 (2021-12-03)
Cite as:
“Galois-qudit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_into_galois