Description
An \(((n,K,d))_q\) Galois-qudit code whose logical subspace is the joint eigenspace of commuting Galois-qudit Pauli operators forming the code's stabilizer group \(\mathsf{S}\). Traditionally, the logical subspace is the joint \(+1\) eigenspace, and the stabilizer group does not contain \(e^{i \phi} I\) for any \(\phi \neq 0\). The distance \(d\) is the minimum weight of a Galois-qudit Pauli string that implements a nontrivial logical operation in the code.
A Galois-qudit stabilizer code encoding an integer number of qudits (\(K=q^k\)) is denoted as \([[n,k]]_q\) or \([[n,k,d]]_q\). This notation differentiates between Galois-qudit and modular-qudit \([[n,k,d]]_{\mathbb{Z}_q}\) stabilizer codes, although the same notation is usually used for both. Galois-qudit stabilizer codes need not encode an integer number of qudits, with \(K=q^{n-\frac{r}{m}}\), where \(r\) is the number of generators of the stabilizer group, and \(q=p^m\) given prime \(p\) for all Galois qudits. As a result, \([[n,k,d]]\) notation is often used with non-integer \(k=\log_q K\).
Galois symplectic representation: The single Galois-qudit Pauli string \(X_{a} Z_{b}\) for \(a,b\in GF(q)\) is converted to the vector \((a|b)\in GF(q)^2\). The multi Galois-qudit version follows naturally.
A pair of Galois-qudit stabilizers on \(n\) Galois qudits with Galois symplectic representation vectors \((a|b)\) and \((a^{\prime}|b^{\prime})\) commute iff their trace symplectic inner product is zero, \begin{align} \text{tr}(a \cdot b^{\prime} - a^{\prime}\cdot b) = \sum_{j=1}^{n} \text{tr}(a_j b^{\prime}_j - a^{\prime}_i b_i) = 0~. \tag*{(1)}\end{align} Galois symplectic representations of stabilizer group elements form a trace-symplectic self-orthogonal linear code over \(GF(q)^{2n}\). The trace-symplectic inner product reduces to the symplectic inner product when the field trace is removed, and a symplectic self-orthogonal set of vectors is automatically trace-symplectic self-orthogonal.
Another correspondence between Galois-qudit Pauli matrices and elements of the Galois field \(GF(q^2)\) yields the one-to-one correspondence between Galois-qudit stabilizer codes and trace-alternating self-orthogonal additive codes over \(GF(q^2)\) [2].
\(GF(q^2)\) representation: An \(n\)-qubit Galois-qudit Pauli stabilizer can be represented as a length-\(n\) vector over \(GF(q^2)\) using the one-to-one correspondence between the \(q^2\) Galois-qudit Pauli matrices and elements of \(GF(q^2)\). Given a basis \((\beta,\beta^q)\) for \(GF(q^2)\) over \(GF(q)\), the vector \((a|b)\in GF(q)^2\) (representing a Galois-qudit Pauli string in the Galois symplectic representation) is in one-to-one correspondence with element \(a \beta + b \beta^q \in GF(q^2)\) [2][3; Thm. 27.3.8].
The sets of \(GF(q^2)\)-represented vectors for all generators yield a trace-alternating self-orthogonal additive code over \(GF(q^2)\).
Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [1]. As such, modular-qudit stabilizer states can be expressed in terms of linear and quadratic functions over \(\mathbb{Z}_p^{mn}\) [4]. Such states correspond to the set of states with positive Wigner functions [5,6].
Galois-qudit stabilizer codes can equivalently [7] (see also [8,9]) be defined using graphs, yielding an analytical form for the codewords [10].
Protection
Detects errors on up to \(d-1\) qudits, and corrects erasure errors on up to \(d-1\) qudits. Corrects errors on \(\left\lfloor (d-1)/2 \right\rfloor\) qudits. There are algorithms to calculate the minimum distance [11]. There are established shortening/lengthening procedures for pure Galois-qudit stabilizer codes [12][2; Table 1].Encoding
Encoder with \(O(n^2)\) gates can be determined in classical runtime of order \(O(n^3)\) [13].Gates
As opposed to modular qudits for composite \(q\), Galois qudits inherit most of the properties of the prime-qudit Clifford group due to the correspondence between a \(q=p^m\) Galois qudit and \(m\) prime qudits of dimension \(p\) [1].Decoding
Syndrome extraction and computation based on classical additive codes [14].Notes
Tables of bounds and examples of Galois-qudit stabilizer codes for various \(n\) and \(k\), based on algorithms developed in Refs. [15,16], are maintained by M. Grassl at this website. A Magma implementation exists at this website. A modular-qudit stabilizer code with composite dimension \(q\) contains a subcode that is isomorphic to a \(p\)-dimensional prime-qudit stabilizer code for every prime factor \(p\) of \(q\), and the distance of the full stabilizer code is upper bound by the distance of this subcode [17].The number of Galois-qudit stabilizer codes was determined in Ref. [5].See Quantum Codes qudit stabilizer database, maintained by N. Aydin, P. Liu, and B. Yoshino [18,19], at this website.Review of nonbinary stabilizer codes [20].Cousins
- Galois-qudit CWS code— Galois-qudit CWS codes whose underlying classical code is a linear \(q\)-ary code are Galois-qudit stabilizer codes containing a cluster-state codeword.
- Modular-qudit stabilizer code— Recalling that \(q=p^m\), Galois-qudit stabilizer codes can also be treated as prime-qudit stabilizer codes on \(mn\) qudits, giving \(k=nm-r\) [1]. The case \(m=1\) reduces to conventional prime-qudit stabilizer codes on \(n\) qudits. A modular-qudit stabilizer code with composite dimension \(q\) contains a subcode that is isomorphic to a \(p\)-dimensional prime-qudit stabilizer code for every prime factor \(p\) of \(q\), and the distance of the full stabilizer code is bounded by the distance of this subcode [17].
- Additive \(q\)-ary code— Galois-qudit stabilizer codes are the closest quantum analogues of additive codes over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
- Dual additive code— Galois-qudit stabilizer codes are in one-to-one correspondence with trace-symplectic self-orthogonal additive codes of length \(2n\) over \(GF(q)\) via the Galois symplectic representation [1]. They are also in one-to-one correspondence with trace-alternating self-orthogonal additive codes of length \(n\) over \(GF(q^2)\) via the \(GF(q^2)\) representation.
- \(t\)-design— Stabilizer states on \(n\) Galois qubits form complex projective 2-designs [21].
- Graph quantum code— Graph quantum codes for \(G=GF(q)\) are a subset of Galois-qudit stabilizer codes [7]. Any Galois-qubit stabilizer code is equivalent to a graph quantum code for \(G=GF(q)\) via a single-Galois-qudit Clifford circuit [7] (see also [8,9]).
- Analog stabilizer code— Prime-qudit stabilizer codes can be converted into analog stabilizer codes whose distance is at least as large as that of the original code [22].
- EA Galois-qudit stabilizer code— EA Galois-qudit stabilizer codes utilize additional ancillary Galois-qudits in a pre-shared entangled state, but reduce to Galois-qudit stabilizer codes when said qudits are interpreted as noiseless physical qudits. Pure Galois-qudit codes can be used to make EA Galois-qudit stabilizer codes [23][24; Thm. 10].
- Subsystem Galois-qudit stabilizer code— Subsystem Galois-qudit stabilizer codes reduce to Galois-qudit stabilizer codes when there are no gauge qudits.
Member of code lists
Primary Hierarchy
References
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- Markus Grassl, private communication, 2024
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- N. Aydin, P. Liu, and B. Yoshino, “Polycyclic Codes Associated with Trinomials: Good Codes and Open Questions”, (2021) arXiv:2106.12065
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- L. G. Gunderman, “Stabilizer Codes with Exotic Local-dimensions”, Quantum 8, 1249 (2024) arXiv:2303.17000 DOI
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- L. Riguang and M. Zhi, “Non-binary Entanglement-assisted Stabilizer Quantum Codes”, (2011) arXiv:1105.5872
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- 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) arXiv:2010.07902 DOI
Page edit log
- Markus Grassl (2024-07-11) — most recent
- Victor V. Albert (2022-07-22)
- Victor V. Albert (2022-04-13)
- Leonid Pryadko (2022-04-13)
- Victor V. Albert (2022-01-12)
- Qingfeng (Kee) Wang (2022-01-07)
Cite as:
“Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/galois_stabilizer