  We consider transfer of single and multi-qubit states on a quasi-1D lattice,
where the time evolutions involved in the state transfer protocol are generated
by only 1D Hamiltonians. We use the quasi-1D isotropic Heisenberg model under a
magnetic field along the $z$ direction, where the spin-spin interaction
strengths along the vertical sublattices, referred to as rungs, are much
stronger than the interactions along other sublattices. Tuning the
field-strength to a special value, in the strong rung-coupling limit, the
quasi-1D isotropic Heisenberg model can be mapped to an effective 1D XXZ model,
where each rung mimics an effective two-level system. Consequently, the
transfer of low-energy rung states from one rung to another can be represented
by a transfer of an arbitrary single-qubit state from one lattice site to
another using the 1D XXZ model. Exploiting this, we propose protocols for
transferring arbitrary single-qubit states from one lattice site to another by
using specific encoding of the single-qubit state into a low-energy rung state,
and a subsequent decoding of the transferred state on the receiver rung. These
encoding and decoding protocols involve a time evolution generated by the 1D
rung Hamiltonian and single-qubit phase gates, ensuring that all
time-evolutions required for transferring the single-qubit state are generated
from 1D Hamiltonians. We show that the performance of the single-qubit state
transfer using the proposed protocol is always better than the same when a
time-evolution generated by the full quasi-1D Hamiltonian is used.
