  We develop a theory of quantum harmonic analysis on lattices in
$\mathbb{R}^{2d}$. Convolutions of a sequence with an operator and of two
operators are defined over a lattice, and using corresponding Fourier
transforms of sequences and operators we develop a version of harmonic analysis
for these objects. We prove analogues of results from classical harmonic
analysis and the quantum harmonic analysis of Werner, including Tauberian
theorems and a Wiener division lemma. Gabor multipliers from time-frequency
analysis are described as convolutions in this setting. The quantum harmonic
analysis is thus a conceptual framework for the study of Gabor multipliers, and
several of the results include results on Gabor multipliers as special cases.
