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 could provide insight into band-structure details of quantum materials that 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 uniform and varying probability density. The latter improves QPI recovery and mitigates Fourier artifacts. We further simplify the motion of the STM tip through an open traveling salesman's problem for greater efficiency and use the tip-path for drift correction. We expect that the implications of our CS approach will be transformative for the exploration of two-dimensional quantum materials.
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