  Trapped-ion has shown great advantages in building quantum computers. While
high fidelity entangling-gate has been realized for few ions, how to maintain
the high fidelity for large scale trapped-ions still remains an open
problem.Here, we present an analysis on arbitrary scale ion chain and focus on
motional-related errors, reported as one of the leading error sources in
state-of-the-art experiments. We theoretically analyze two-qubit
entangling-gate infidelity in a large ion crystal. To verify our result, we
develop an efficient numerical simulation algorithm that avoids exponential
increases of the Hilbert space dimension. For the motional heating error, We
derive a much tighter bound of gate infidelity than previously estimated
$O(N\Gamma\tau)$, and we give an intuitive understanding from the trajectories
in the phase space of motional modes. Our discoveries may inspire the scheme of
pulse design against incoherent errors and shed light on the way toward
constructing scalable quantum computers with large ion crystals.
