From b27eafac6b8dc96d7133e85be690ffebf56e6b8f Mon Sep 17 00:00:00 2001 From: VVA2024 Date: Tue, 27 Aug 2024 18:04:31 -0400 Subject: [PATCH] ~ --- codes/classical/bits/ta-shma.yml | 2 +- codes/quantum/properties/block/block_quantum.yml | 1 + .../topological/surface/hyperbolic/freedman_meyer_luo.yml | 7 ++++++- 3 files changed, 8 insertions(+), 2 deletions(-) diff --git a/codes/classical/bits/ta-shma.yml b/codes/classical/bits/ta-shma.yml index 09759afaf..75abc1f84 100644 --- a/codes/classical/bits/ta-shma.yml +++ b/codes/classical/bits/ta-shma.yml @@ -11,7 +11,7 @@ name: 'Ta-Shma zigzag code' introduced: '\cite{doi:10.1145/3055399.3055408}' description: | - Member of a family of \(\epsilon\)-balanced codes that nearly achieves the \hyperref[topic:gv-bound]{asymptotic GV bound}. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of order \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) \cite{arxiv:2011.05500}. + Member of a family of \(\epsilon\)-balanced codes that nearly achieves the \hyperref[topic:gv-bound]{asymptotic GV bound}. The codes have relative distance \(\frac{1}{2}-\frac{\epsilon}{2}\) and rate of \hyperref[topic:asymptotics]{order} \(\Omega (\epsilon^{2+\beta})\) for \(\beta\to 0\) as \(n\to\infty\) \cite{arxiv:2011.05500}. features: decoders: diff --git a/codes/quantum/properties/block/block_quantum.yml b/codes/quantum/properties/block/block_quantum.yml index 60d8bf48a..da3d30b27 100644 --- a/codes/quantum/properties/block/block_quantum.yml +++ b/codes/quantum/properties/block/block_quantum.yml @@ -26,6 +26,7 @@ protection: | \subsection{Bounds on code parameters} Bounds on finite dimensional block code performance include the quantum Singleton bound, quantum Hamming bound, \hyperref[topic:quantum-gv-bound]{quantum GV bound}, various quantum linear programming (LP) bounds \cite{arxiv:quant-ph/9611001,arxiv:quant-ph/9709049} (see the book \cite{preset:GottesmanBook}), and other bounds \cite{doi:10.1109/TIT.2005.862086,arxiv:1007.3655}. A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code). + We are often interested in how parameters of particular infinite block quanutm code families scale with increasing block length \(n\), necessitating the use of \hyperref[topic:asymptotics]{asymptotic notation}. \begin{defterm}{Quantum GV bound} \label{topic:quantum-gv-bound} diff --git a/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml b/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml index 2d2a2ebf0..eb9512532 100644 --- a/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml +++ b/codes/quantum/qubits/stabilizer/topological/surface/hyperbolic/freedman_meyer_luo.yml @@ -11,13 +11,18 @@ name: 'Freedman-Meyer-Luo code' introduced: '\cite{doi:10.1201/9781420035377-13}' description: | - Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom \cite{doi:10.2140/gtm.1999.2.113}. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries \cite{doi:10.1063/1.4726034}. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology. + Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting systolic freedom \cite{doi:10.2140/gtm.1999.2.113}. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries \cite{doi:10.1063/1.4726034}. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology. + + The Freedman-Meter-Luo code has been generalized to a family with rate of \hyperref[topic:asymptotics]{order} \(O(1/\sqrt{\log n})\) and minimum distance of \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates \cite{arxiv:2310.16982}. protection: 'Four-dimensional manifolds with weak systolic freedom yield \([[n,2,\Omega(\sqrt{n \sqrt{\log n}})]]\) surface codes.' features: rate: 'Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes.' + fault_tolerance: + - 'The Freedman-Meter-Luo code has been generalized to a family with rate of \hyperref[topic:asymptotics]{order} \(O(1/\sqrt{\log n})\) and minimum distance of \hyperref[topic:asymptotics]{order} \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates \cite{arxiv:2310.16982}.' + notes: - 'See thesis by Fetaya for pedagogical exposition \cite{arxiv:1108.2886}.'