  Scanning tunneling microscopy (STM) is a notoriously slow technique;
Data-recording is serial which renders complex measurement tasks, such as
quasiparticle interference (QPI) mapping, impractical. However, QPI would
provide insight into band-structure details of quantum materials which can be
inaccessible to angle-resolved photoemission spectroscopy. Here we use
compressed sensing (CS) to fundamentally speed-up QPI mapping. We reliably
recover the QPI information from a fraction of the usual local density of state
measurements. The requirement of CS is naturally fulfilled for QPI, since CS
relies on sparsity in a vector domain, here given by few nonzero coefficients
in Fourier space. We exemplify CS on a simulated Cu(111) surface using random
sampling of constant and varying probability density. We further simplify the
motion of the STM tip through an open traveling salesman's problem for greater
efficiency. We expect that the implications of our CS approach will be
transformative for the exploration of 2D quantum materials.
