Error- and Loss-Tolerances of Surface Codes with General Lattice Structures

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Here, the logical error probabilities p Z , X l p^{l}_{Z,X} mean that after the recovery operations logical Z Z and X X operators act on the code space with probabilities p Z , X l p^{l}_{Z,X} , respectively. If a logical Z Z ( X X ) error occurs, the logical X X ( Z Z ) information is flipped, since the logical operators are anticommute. If p < p Z th p<p_{Z}^{\rm th} ( p < p X th p<p_{X}^{\rm th} ), the probability p Z l p^{l}_{Z} ( p X l p^{l}_{X} ) of such a logical Z Z ( X X ) error is suppressed rapidly by increasing the lattice size L L

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The numerical simulations are performed on the L × L L\times L lattice with L = 16 , 24 , 32 L=16,24,32 for the square lattice. For other structures, the lattices of the similar numbers of qubits are used

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It is, of course, possible to construct surface codes, which are robust against X X errors, by defining their stabilizer operators on the duals of the Kagome, hexagonal, and tri-hexa lattices as shown in Fig. 2

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When p loss p^{\rm loss} is near the bond percolation threshold, the numerically obtained p th p^{\rm th} lies below the drawn curve (not shown in Fig. 3

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This provides an intuitive understanding of difficulty to observe superposition of macroscopically distinct two quantum states, i.e., the Schrödinger’s cat paradox

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The preliminary version of this work was presented at QIP 2012 conference held in Dec. 2011

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