  We propose the use of 2-dimensional Penning trap arrays as a scalable
platform for quantum simulation and quantum computing with trapped atomic ions.
This approach involves placing arrays of micro-structured electrodes defining
static electric quadrupole sites in a magnetic field, with single ions trapped
at each site and coupled to neighbors via the Coulomb interaction. We solve for
the normal modes of ion motion in such arrays, and derive a generalized
multi-ion invariance theorem for stable motion even in the presence of trap
imperfections. We use these techniques to investigate the feasibility of
quantum simulation and quantum computation in fixed ion lattices. In
homogeneous arrays, we show that sufficiently dense arrays are achievable, with
axial, magnetron and cyclotron motions exhibiting inter-ion dipolar coupling
with rates significantly higher than expected decoherence. With the addition of
laser fields these can realize tunable-range interacting spin Hamiltonians. We
also show how local control of potentials allows isolation of small numbers of
ions in a fixed array and can be used to implement high fidelity gates. The use
of static trapping fields means that our approach is not limited by power
requirements as system size increases, removing a major challenge for scaling
which is present in standard radio-frequency traps. Thus the architecture and
methods provided here appear to open a path for trapped-ion quantum computing
to reach fault-tolerant scale devices.
