  The problem of sharing entanglement over large distances is crucial for
implementations of quantum cryptography. A possible scheme for long-distance
entanglement sharing and quantum communication exploits networks whose nodes
share Einstein-Podolsky-Rosen (EPR) pairs. In Perseguers et al. [Phys. Rev. A
78, 062324 (2008)] the authors put forward an important isomorphism between
storing quantum information in a dimension $D$ and transmission of quantum
information in a $D+1$-dimensional network. We show that it is possible to
obtain long-distance entanglement in a noisy two-dimensional (2D) network, even
when taking into account that encoding and decoding of a state is exposed to an
error. For 3D networks we propose a simple encoding and decoding scheme based
solely on syndrome measurements on 2D Kitaev topological quantum memory. Our
procedure constitutes an alternative scheme of state injection that can be used
for universal quantum computation on 2D Kitaev code. It is shown that the
encoding scheme is equivalent to teleporting the state, from a specific node
into a whole two-dimensional network, through some virtual EPR pair existing
within the rest of network qubits. We present an analytic lower bound on
fidelity of the encoding and decoding procedure, using as our main tool a
modified metric on space-time lattice, deviating from a taxicab metric at the
first and the last time slices.

Quantum Physics