  Encoding logical quantum information in harmonic oscillator modes is a
promising and hardware-efficient approach to the realization of a quantum
computer. In this work, we propose to encode logical qubits in grid states of
an ensemble of harmonic oscillator modes. We first discuss general results
about these multimode bosonic codes; how to design them, how to practically
implement them in different experimental platforms and how lattice symmetries
can be leveraged to perform logical non-Clifford operations. We then introduce
in detail two two-mode grid codes based on the hypercubic and D4 lattices,
respectively, showing how to perform a universal set of logical operations. We
demonstrate numerically that multimode grid codes have, compared to their
single-mode counterpart, increased robustness against propagation of errors
from ancillas used for error correction. Finally, we highlight some interesting
links between multidimensional lattices and single-mode grid codes concatenated
with qubit codes.
