We developed a targeted chromosome conformation capture (4C) approach that uses unique molecular identifiers (UMIs) to derive high-complexity quantitative chromosome contact profiles with controlled signal-to-noise ratios. UMI-4C detects chromosomal interactions with improved sensitivity and specificity, and it can easily be multiplexed to allow robust comparison of contact distributions between loci and conditions. This approach may open the way to the incorporation of contact distributions into quantitative models of gene regulation.