Galois-qudit quantum RM code[1] 

Description

True \(q\)-Galois-qudit stabilizer code constructed from generalized Reed-Muller (GRM) codes via either the Hermitian construction or the Galois-qudit CSS construction.

The CSS construction yields a \([[q^m,k(v_2)-k(v_1),\min\{d(v_1^{\perp}),d(v_2)\}]]_q\) code, constructed from the generalized Reed Muller Codes RM\(_q(v_1,m)\) and RM\(_q(v_2,m)\), with \(0\leq v_1 \leq v_2 \leq q(m-1)-1\). The parameters are \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq}{v-jq}\) and \(d(v) = (R+1)q^{Q}\), where \(m(q-1)-v=(q-1)Q+R\) so that \(0\leq R\leq q-1\). Here \(0\leq v_1,v_2 \leq m(q-1)-1\), \(q\) is prime, and \(m\) is a positive integer.

Using the code GRM\(_{q^2}(v,m)\) for \(0\leq v \leq q(m-1)-1\), the Hermitian construction yields a pure \([[q^{2m},q^{2m}-k(v),d(v^{\perp})]]_q\) quantum code where \(k(v) = \sum_{j=0}^{m}(-1)^{j}\dbinom{m}{j}\dbinom{m+v-jq^2}{v-jq^2}\) and \(d(v^{\perp}) = (R+1)q^{2Q}\) with \(v+1 = (q^2 - 1)Q + R\).

For a CSS code constructed from classical codes \(C_1\) and \(C_2\), the punctured code is defined as the classical code \(P(C) = \{(a_ib_i)_{i=1}^{n}|a \in C_1, b \in C_2^{\perp}\}^{\perp}\). Quantum RM codes can be punctured to any length \(r\), provided \(P(C) = \mathcal{R}_q(v_2-v_1,m)\) has a codeword of this weight.

Protection

QRM\(_{d}(m)\) quantum codes are \(\mathcal{M}_{d}^{m}\) distillation codes of distance \(D=2\). We define a \(\mathcal{M}_{d}^{m}\) distillation code as any \(n\) Galois-qudit stabilizer code \(C\) having the following properties: (a) All \(M \in \mathcal{M}_{d}^{m}\) are transversal so that \(M^{\otimes n}C(M^{\otimes n})^{\dagger} = M_{L}^{\dagger}CM_{L}\), (b) the code has distance \(D \geq 2\), and (c) the code has logical pauli operators \(X_{L} = X[\mathbf{1}]\) and \(Z_{L} = Z[(d-1)\mathbf{1}]\). Here, \(\mathbf{1}\) is a shorthand for the vector \((1,1, \dots, 1)\).

Transversal Gates

QRM\(_{d}(m)\) quantum codes are \(\mathcal{M}_{d}^{m}\) distillation codes of distance \(D=2\).

Parent

Child

Cousins

References

[1]
P. K. Sarvepalli and A. Klappenecker, “Nonbinary Quantum Reed-Muller Codes”, (2005) arXiv:quant-ph/0502001
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Zoo Code ID: galois_reed_muller

Cite as:
“Galois-qudit quantum RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/galois_reed_muller
BibTeX:
@incollection{eczoo_galois_reed_muller, title={Galois-qudit quantum RM code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/galois_reed_muller} }
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Cite as:

“Galois-qudit quantum RM code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/galois_reed_muller

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits_galois/evaluation/galois_reed_muller.yml.