  We report on the design of a cryogenic setup for trapped ion quantum
computing containing a segmented surface electrode trap. The heat shield of our
cryostat is designed to attenuate alternating magnetic field noise, resulting
in 120~dB reduction of 50~Hz noise along the magnetic field axis. We combine
this efficient magnetic shielding with high optical access required for single
ion addressing as well as for efficient state detection by placing two lenses
each with numerical aperture 0.23 inside the inner heat shield. The cryostat
design incorporates vibration isolation to avoid decoherence of optical qubits
due to the motion of the cryostat. We measure vibrations of the cryostat of
less than $\pm$20~nm over 2~s. In addition to the cryogenic apparatus, we
describe the setup required for an operation with
$^{\mathrm{40}}$Ca$^{\mathrm{+}}$ and $^{\mathrm{88}}$Sr$^{\mathrm{+}}$ ions.
The instability of the laser manipulating the optical qubits in
$^{\mathrm{40}}$Ca$^{\mathrm{+}}$ is characterized yielding a minimum of its
Allan deviation of 2.4$\cdot$10$^{\mathrm{-15}}$ at 0.33~s. To evaluate the
performance of the apparatus, we trapped $^{\mathrm{40}}$Ca$^{\mathrm{+}}$
ions, obtaining a heating rate of 2.14(16)~phonons/s and a Gaussian decay of
the Ramsey contrast with a 1/e-time of 18.2(8)~ms.
