Galois-qudit stabilizer code[1][2]

Description

An \(((n,K,d))_{GF(q)}\) 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 \([[n,k]]_{GF(q)}\) or \([[n,k,d]]_{GF(q)}\). This notation differentiates between Galois-qudit and modular-qudit stabilizer codes, although the same notation, \([[n,k,d]]_q\), 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 \(k=\log_q K\).

The stabilizer commutation condition can equivalently be stated in the symplectic representation. A pair of Galois-qudit stabilizers on \(n\) Galois qudits with 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~. \end{align} Symplectic representations of stabilizer group elements thus form a self-orthogonal subspace of \(GF(q)^{2n}\) with respect to the trace-symplectic inner product.

Note that the above trace-symplectic inner product reduces to the symplectic inner product when the trace is removed, and a symplectic self-orthogonal set of vectors is automatically trace-symplectic self-orthogonal. More generally, any additive classical code whose self-orthogonality under some inner product (such as Hermitian, Euclidean, or symplectic) implies trace-symplectic self-orthogonality of an equivalent code can be used to construct a Galois-qudit stabilizer code (see children).

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.

Encoding

Encoder with \(O(n^2)\) gates can be determined in classical runtime of order \(O(n^3)\) [3].

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].

Notes

The number of Galois-qudit stabilizer codes was determined in Ref. [4].

Parents

  • Stabilizer code
  • Galois-qudit non-stabilizer code — A non-stabilizer code is also a stabilizer code if its Fourier description \(\mathsf{B}\) is a subgroup of some Gottesman subgroup \(\mathsf{S}\). When \(\mathsf{B}\) is just a subset, the code is explicitly not a stabilizer code.

Child

Cousins

  • Qubit stabilizer code — Galois-qudit stabilizer codes reduce to qubit stabilizer codes for \(q=2\).
  • 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.
  • Additive \(q\)-ary code — A Galois-qudit stabilizer code is the closest quantum analogue of an additive code over \(GF(q)\) because addition in the field corresponds to multiplication of stabilizers in the quantum case.
  • Dual additive code — An additive code of length \(2n\) over \(GF(q)\) that is self-orthogonal with respect to the trace-symplectic inner product corresponds to symplectic representations of an \(n\) Galois-qudit stabilizer group [1]. Moreover, any additive code whose self-orthogonality under some inner product (such as Hermitian, Euclidean, or symplectic) implies trace-symplectic self-orthogonality of an equivalent code can be used to construct a Galois-qudit stabilizer code.

References

[1]
A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001). DOI
[2]
Avanti Ketkar et al., “Nonbinary stabilizer codes over finite fields”. quant-ph/0508070
[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). DOI; quant-ph/0211014
[4]
D. Gross, “Hudson’s theorem for finite-dimensional quantum systems”, Journal of Mathematical Physics 47, 122107 (2006). DOI; quant-ph/0602001
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Internal code ID: galois_stabilizer

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Zoo Code ID: galois_stabilizer

Cite as:
“Galois-qudit stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/galois_stabilizer
BibTeX:
@incollection{eczoo_galois_stabilizer, title={Galois-qudit stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_stabilizer} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits_galois/galois_stabilizer.yml.