  Penning traps have been used for performing quantum simulations and sensing
with hundreds of ions and provide a promising route toward scaling up trapped
ion quantum platforms because of the ability to trap and control up to
thousands of ions in 2D and 3D crystals. A leading source of decoherence in
laser-based multiqubit operations on trapped ions is off-resonant spontaneous
emission. While many trapped ion quantum computers or simulators utilize clock
qubits, other systems rely on Zeeman qubits, which require a more complex
calculation of this decoherence. We examine theoretically the impacts of
spontaneous emission on quantum gates performed with trapped ions in a high
magnetic field. We consider two types of gates -- light-shift and
Molmer-Sorensen gates -- and compare the decoherence errors in each. We also
compare different detunings, polarizations, and required intensities of the
laser beams used to drive the gates. We show that both gates can have similar
performance at their optimal operating conditions and examine the experimental
feasibility of various operating points. By examining the magnetic field
dependence of each gate, we demonstrate that when the $P$ state fine structure
splitting is large compared to the Zeeman splittings, the theoretical
performance of the Molmer-Sorensen gate is significantly better than that of
the light-shift gate. Additionally, for the light-shift gate, we make an
approximate comparison between the fidelities that can be achieved at high
fields with the fidelities of state-of-the-art two-qubit trapped ion quantum
gates. We show that, with regard to spontaneous emission, the achievable
infidelity of our current configuration is about an order of magnitude larger
than that of the best low-field gates, but we also discuss alternative
configurations with potential error rates that are comparable with
state-of-the-art trapped ion gates.
