  We demonstrate parallel composite quantum logic gates with phases implemented
locally through nanoscale movement of ions within a global laser beam of fixed
pulse duration. We show that a simple four-pulse sequence suffices for
constructing ideal arbitrary single-qubit rotations in the presence of large
intensity inhomogeneities across the ion trap due to laser beam-pointing or
beam-focusing. Using such sequences, we perform parallel arbitrary rotations on
ions in two trapping zones separated by 700 $\mu$m with fidelities comparable
to those of our standard laser-controlled gates. Our scheme improves on current
transport or zone-dependent quantum gates to include phase modulation with
local control of the ion's confinement potential. This enables a scalable
implementation of an arbitrary number of parallel operations on densely packed
qubits with a single laser modulator and beam path.

Quantum Physics;Atomic Physics;Optics