  Quantum computers are expected to achieve a significant speed-up over
classical computers in solving a range of computational problems. Chains of
ions held in a linear Paul trap are a promising platform for constructing such
quantum computers, due to their long coherence times and high quality of
control. Here we report on the construction of a small, five-qubit, universal
quantum computer using $^{88}\text{Sr}^{+}$ ions in an RF trap. All basic
operations, including initialization, quantum logic operations, and readout,
are performed with high fidelity. Selective two-qubit and single-qubit gates,
implemented using a narrow linewidth laser, comprise a universal gate set,
allowing realization of any unitary on the quantum register. We review the main
experimental tools, and describe in detail unique aspects of the computer: the
use of robust entangling gates and the development of a quantum coherent
feedback system through EMCCD camera acquisition. The latter is necessary for
carrying out quantum error correction protocols in future experiments.
