diff --git a/codes/classical/analog/analog.yml b/codes/classical/analog/analog.yml index ae7b9e8d1..63550e8d8 100644 --- a/codes/classical/analog/analog.yml +++ b/codes/classical/analog/analog.yml @@ -33,7 +33,7 @@ protection: | Defining a density for infinite constellations can be done using a limit \cite[pg. 349]{preset:EricZin}. The Kabatiansky-Levenshtein bound \cite{manual:{G. A. Kabatiansky, V. I. Levenshtein, “On Bounds for Packings on a Sphere and in Space”, Probl. Peredachi Inf., 14:1 (1978), 3–25; Problems Inform. Transmission, 14:1 (1978), 1–17}} says that any sphere packing must satistfy \(\frac{1}{n}\log_{2}\Delta\lesssim-0.5990\) asymptotically with dimension \(n\). - Other bounds include the Rogers bound \cite{doi:10.1112/plms/s3-8.4.609}. + Other bounds include the Rogers bound \cite{doi:10.1112/plms/s3-8.4.609} and its recent improvement \cite{arxiv:2312.10026}. For more details, see \cite[Ch. 10.4]{preset:EricZin}\cite[Ch. 1]{doi:10.1007/978-1-4757-6568-7}. The \textit{covering problem} asks how one can cover all of space by overlapping spheres in the most efficient way. diff --git a/codes/quantum/qubits/small_distance/quantum_hamming.yml b/codes/quantum/qubits/small_distance/quantum_hamming.yml index 4454d5f64..d6dc23dd3 100644 --- a/codes/quantum/qubits/small_distance/quantum_hamming.yml +++ b/codes/quantum/qubits/small_distance/quantum_hamming.yml @@ -7,14 +7,16 @@ code_id: quantum_hamming physical: qubits logical: qubits -name: 'Gottesman code' +name: '\([[2^r, 2^r-r-2, 3]]\) Gottesman code' short_name: '\([[2^r, 2^r-r-2, 3]]\)' introduced: '\cite{arXiv:quant-ph/9604038}' alternative_names: - '\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code' -description: 'A family of stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. Can be obtained from the CSS construction with a \([2^r,r+1,2^{r-1}] = C_2^{\perp}\) RM code and a \([2^r,2^r-1,2] = C_1\) even-weight code \cite{arxiv:quant-ph/9605021}.' +description: 'A family of non-CSS stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound.' + +# Can be obtained from the CSS construction with a \([2^r,r+1,2^{r-1}] = C_2^{\perp}\) RM code and a \([2^r,2^r-1,2] = C_1\) even-weight code \cite{arxiv:quant-ph/9605021}. protection: 'Protects against any single qubit error.' @@ -25,11 +27,12 @@ relations: - code_id: small_distance_quantum cousins: - code_id: quantum_perfect - detail: 'Quantum Hamming codes saturate the asymptotic quantum Hamming bound.' + detail: '\([[2^r, 2^r-r-2, 3]]\) Gottesman codes saturate the asymptotic quantum Hamming bound.' - code_id: hamming - detail: '\([[2^r, 2^r-r-2, 3]]\) quantum Hamming codes are analogues of Hamming codes in that they saturate the asymptotic Hamming bound.' - - code_id: reed_muller - detail: '\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code can be obtained from the CSS construction using a first-order \([2^r,r+1,2^{r-1}]\) RM code and a \([2^r,2^r-1,2]\) even-weight code \cite{arxiv:quant-ph/9605021}.' + detail: '\([[2^r, 2^r-r-2, 3]]\) Gottesman codes are analogues of Hamming codes in that they saturate the asymptotic Hamming bound.' + +# - code_id: reed_muller +# detail: 'Gottesman codes can be obtained from the CSS construction using a first-order \([2^r,r+1,2^{r-1}]\) RM code and a \([2^r,2^r-1,2]\) even-weight code \cite{arxiv:quant-ph/9605021}.' # Begin Entry Meta Information