The problem of sharing entanglement over large distances is crucial forimplementations of quantum cryptography. A possible scheme for long-distanceentanglement sharing and quantum communication exploits networks whose nodesshare Einstein-Podolsky-Rosen (EPR) pairs. In Perseguers et al. [Phys. Rev. A78, 062324 (2008)] the authors put forward an important isomorphism betweenstoring quantum information in a dimension $D$ and transmission of quantuminformation in a $D+1$-dimensional network. We show that it is possible toobtain long-distance entanglement in a noisy two-dimensional (2D) network, evenwhen taking into account that encoding and decoding of a state is exposed to anerror. For 3D networks we propose a simple encoding and decoding scheme basedsolely on syndrome measurements on 2D Kitaev topological quantum memory. Ourprocedure constitutes an alternative scheme of state injection that can be usedfor universal quantum computation on 2D Kitaev code. It is shown that theencoding scheme is equivalent to teleporting the state, from a specific nodeinto a whole two-dimensional network, through some virtual EPR pair existingwithin the rest of network qubits. We present an analytic lower bound onfidelity of the encoding and decoding procedure, using as our main tool amodified metric on space-time lattice, deviating from a taxicab metric at thefirst and the last time slices.
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