  In this work we address the problem of realizing a reliable quantum memory
based on zero-energy Majorana modes in the presence of experimental constraints
on the operations aimed at recovering the information. In particular, we
characterize the best recovery operation acting only on the zero-energy
Majorana modes and the memory fidelity that can be therewith achieved. In order
to understand the effect of such restriction, we discuss two examples of noise
models acting on the topological system and compare the amount of information
that can be recovered by accessing either the whole system, or the zero-modes
only, with particular attention to the scaling with the size of the system and
the energy gap. We explicitly discuss the case of a thermal bosonic environment
inducing a parity-preserving Markovian dynamics in which the introduced memory
fidelity decays exponentially in time, independent from system size, thus
showing the impossibility to retrieve the information by acting on the
zero-modes only. We argue, however, that even in the presence of experimental
limitations, the Hamiltonian gap is still beneficial to the storage of
information.
