Colloidal quantum dots (QDs) are attractive materials for realizing solution-processable laser diodes that could benefit from size-controlled emission wavelengths, low optical-gain thresholds and ease of integration with photonic and electronic circuits^{1,2,3,4,5,6,7}. However, the implementation of such devices has been hampered by fast Auger recombination of gain-active multicarrier states^1,8, poor stability of QD films at high current densities^9,10 and the difficulty to obtain net optical gain in a complex device stack wherein a thin electroluminescent QD layer is combined with optically lossy charge-conducting layers^11,12,13. Here we resolve these challenges and achieve amplified spontaneous emission (ASE) from electrically pumped colloidal QDs. The developed devices use compact, continuously graded QDs with suppressed Auger recombination incorporated into a pulsed, high-current-density charge-injection structure supplemented by a low-loss photonic waveguide. These colloidal QD ASE diodes exhibit strong, broadband optical gain and demonstrate bright edge emission with instantaneous power of up to 170 μW.