diff --git a/codes/quantum/properties/block/quantum_mds.yml b/codes/quantum/properties/block/quantum_mds.yml index 7653641ea..a2a61527d 100644 --- a/codes/quantum/properties/block/quantum_mds.yml +++ b/codes/quantum/properties/block/quantum_mds.yml @@ -9,15 +9,23 @@ code_id: quantum_mds name: 'Quantum maximum-distance-separable (MDS) code' short_name: 'Quantum MDS' -introduced: '\cite{arxiv:quant-ph/9702031}' +introduced: '\cite{arxiv:quant-ph/9702031,arxiv:quant-ph/9703048}' description: | - An \(((n,q^k,d))\) code constructed out of \(q\)-dimensional qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound + An \(((n,K,d))\) code constructed out of \(q\)-dimensional qubits or Galois qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound \begin{align} - 2(d-1) \leq n-k + K \leq q^{n-2(d-1)} \end{align} becomes an equality. - Such codes are pure \cite{arxiv:1907.07733}. + When \(K = q^k\) for some integer \(k\), the above reduces to \(2(d-1) \leq n-k\). + Such codes are pure \cite{arxiv:quant-ph/9703048}. + +# An \(((n,q^k,d))\) code constructed out of \(q\)-dimensional qudits is an MDS code if parameters \(n\), \(k\), \(d\), and \(q\) are such that the quantum Singleton bound +# \begin{align} +# 2(d-1) \leq n-k +# \end{align} +# becomes an equality. +# Such codes are pure \cite{arxiv:quant-ph/9703048}. protection: 'Given \(n\) and \(k\), MDS codes have the highest distance possible of all codes and so have the best possible error correction properties.' diff --git a/codes/quantum/qubits/small_distance/quantum_hamming.yml b/codes/quantum/qubits/small_distance/quantum_hamming.yml index e40955e6d..4454d5f64 100644 --- a/codes/quantum/qubits/small_distance/quantum_hamming.yml +++ b/codes/quantum/qubits/small_distance/quantum_hamming.yml @@ -7,11 +7,14 @@ code_id: quantum_hamming physical: qubits logical: qubits -name: '\([[2^r, 2^r-r-2, 3]]\) quantum Hamming code' +name: 'Gottesman code' short_name: '\([[2^r, 2^r-r-2, 3]]\)' introduced: '\cite{arXiv:quant-ph/9604038}' -description: 'A family of stabilizer codes of distance \(3\) that saturate the asymptotic quantum Hamming bound. 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}.' +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}.' protection: 'Protects against any single qubit error.' diff --git a/codes/quantum/qubits/stabilizer/qubit_css.yml b/codes/quantum/qubits/stabilizer/qubit_css.yml index e73fb824a..a32cfe72c 100644 --- a/codes/quantum/qubits/stabilizer/qubit_css.yml +++ b/codes/quantum/qubits/stabilizer/qubit_css.yml @@ -92,6 +92,8 @@ protection: | The distance of the CSS code is equal to the minimum of the combinatorial (\(d-1\))-systole of the cellulated \(d\)-dimensional manifold and its dual. + CSS codes have a \textit{CSS lower bound} against depolarizing noise, quantified by lower bounds on independently decoding the two classical codes \cite{doi:10.1109/ISIT.2013.6620358}. + features: rate: 'For a depolarizing channel with probability \(p\), CSS codes allowing for arbitrarily accurate recovery exist with asymptotic rate \(1-2h(p)\), where \(h\) is the binary entropy function \cite{arxiv:quant-ph/0110143}.' encoders: diff --git a/codes/quantum/qubits/stabilizer/topological/surface/higher_dim_surface/3d_surface.yml b/codes/quantum/qubits/stabilizer/topological/surface/higher_dim_surface/3d_surface.yml index a988b1278..54c25a9ea 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/higher_dim_surface/3d_surface.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/higher_dim_surface/3d_surface.yml @@ -22,6 +22,8 @@ description: | The construction on surfaces with boundaries is often called the \textit{3D surface code}. + Related models \cite{arxiv:cond-mat/0607736,arxiv:1012.0859} on lattices with certain colorability are equivalent to several copies of the 3D surface code \cite{arXiv:1908.08049}. + protection: | The planar 3D surface code family on a cubic lattice of length \(L\) has parameters \([[2L(L-1)^2+L^3,1,d_X=L^2,d_Z=L]]\), while the 3D toric code has parameters \([[3L^3,3,d_X=L^2,d_Z=L]]\). diff --git a/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml b/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml index 60207a812..30ae9313e 100644 --- a/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml +++ b/codes/quantum/qudits_galois/stabilizer/stabilizer_over_gfqsq.yml @@ -8,7 +8,7 @@ physical: galois logical: galois name: 'Hermitian-construction code' -introduced: '\cite{arXiv:quant-ph/0508070}' +introduced: '\cite{doi:10.1109/18.959288,arXiv:quant-ph/0508070}' # alternative_names: # - 'Stabilizer code over \(GF(q^2)\)'