  Trapped-ion quantum computers have demonstrated high-performance gate
operations in registers of about ten qubits. However, scaling up and
parallelizing quantum computations with long one-dimensional (1D) ion strings
is an outstanding challenge due to the global nature of the motional modes of
the ions which mediate qubit-qubit couplings. Here, we devise methods to
implement scalable and parallel entangling gates by using engineered localized
phonon modes. We propose to tailor such localized modes by tuning the local
potential of individual ions with programmable optical tweezers. Localized
modes of small subsets of qubits form the basis to perform entangling gates on
these subsets in parallel. We demonstrate the inherent scalability of this
approach by presenting analytical and numerical results for long 1D ion chains
and even for infinite chains of uniformly spaced ions. Furthermore, we show
that combining our methods with optimal coherent control techniques allows to
realize maximally dense universal parallelized quantum circuits.
