  Measurements on cluster states can be used to process quantum information.
But errors in cluster states naturally accrue as error-prone inter-particle
interactions entangle qubits. We consider one-dimensional cluster states built
from controlled phase, Ising, and XY interactions with slow two-qubit error in
the interaction strength, consistent with error models of interactions found in
a variety of qubit architectures. We focus on measurement protocols designed to
implement perfect teleportation wherein quantum information moves across a
cluster state intact. Deviations from perfect teleportation offer a proxy for
entanglement that can be degraded by two-qubit gate errors. We detail an
experimentally viable teleportation fidelity that offers a measure of the
impact of error on the cluster state as a whole. Our fidelity calculations show
that the error has a distinctly different impact depending on the underlying
interaction used for the two-qubit entangling gate. In particular, the Ising
and XY interactions can allow perfect teleportation through the cluster state
even with large errors, but the controlled phase interaction does not.
Nonetheless, we find that teleportation through cluster state chains of size
$N$ has a maximum two-qubit error for teleportation along a quantum channel
that decreases as $N^{-1/2}$. To allow construction of larger cluster states,
we also design lowest-order refocusing pulses for correcting slow errors in the
interaction strength. Our work generalizes to higher-dimensional cluster states
and sets the stage for experiments to monitor the growth of entanglement in
cluster states built from error-prone interactions.
