Merge branch 'main' of http://github.com/errorcorrectionzoo/eczoo_data

Author: valbert4

codes/quantum/qubits/stabilizer/topological/surface/2d_surface/toric/toric.yml

@@ -56,6 +56,7 @@ features:
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       Independent \(X,Z\) noise: \(p_X = 10.31\%\) under MWPM decoding \cite{arxiv:quant-ph/0207088} (see also Ref. \cite{arxiv:1405.4883}), \(9.9\%\) under BP-OSD decoding \cite{arxiv:2005.07016}, and \(8.9\%\) under GBP decoding \cite{arxiv:2212.03214}. 
       The threshold under ML decoding corresponds to the value of a critical point of a two-dimensional random-bond Ising model (RBIM) on the Nishimori line \cite{doi:10.1143/JPSJ.55.3305,arxiv:quant-ph/0110143}, calculated to be \(10.94 \pm 0.02\%\) in Ref. \cite{arxiv:cond-mat/0010143}, \(10.93(2)\%\) in Ref. \cite{arxiv:cond-mat/0106023}, \(10.9187\%\) in Ref. \cite{arxiv:0811.0464}, \(10.917(3)\%\) in Ref. \cite{arxiv:0811.2101}, \(10.939(6)\%\) in Ref. \cite{arxiv:0902.4153}, and estimated to be between \(10.9\%\) and \(11\%\) in Ref. \cite{arxiv:1405.4883}.
+      The model for the case of the surface code has been thoroughly investigated \cite{arxiv:2512.10399}.
       The Bravyi-Suchara-Vargo (BSV) tensor network decoder \cite{arxiv:1405.4883} exactly solves the ML decoding problem under independent \(X,Z\) noise for the surface code and has complexity of \hyperref[topic:asymptotics]{order} \(O(n^2)\); the decoder provides an efficient tensor-network contraction for the partition function resulting from the statistical mechanical mapping, which is known to be solvable for an Ising model on a planar graph \cite{doi:10.1103/PhysRev.88.1332}.
       ML decoding \cite{arxiv:quant-ph/0110143} is \(\#P\)-hard in general for the surface code \cite{arxiv:2309.10331}.
       Above values are for one type of noise only, and the ML threshold for combined \(X\) and \(Z\) noise is \(2p_X - p_X^2 \approx 20.6\%\) \cite[Table 1]{arxiv:2212.03214}.