We present a derivation of the star formation rate per comoving volume of quasar host galaxies, derived from stacking analyses of far-infrared to mm-wave photometry of quasars with redshifts 0 < z < 6 and absolute I-band magnitudes -22 > IAB > -32 We use the science demonstration observations of the first ~16 deg2 from the Herschel Astrophysical Terahertz Large Area Survey (H-ATLAS) in which there are 240 quasars from the Sloan Digital Sky Survey (SDSS) and a further 171 from the 2dF-SDSS LRG and QSO (2SLAQ) survey. We supplement this data with a compilation of data from IRAS, ISO, Spitzer, SCUBA and MAMBO. H-ATLAS alone statistically detects the quasars in its survey area at >5σ at 250,350 and 500 μm. From the compilation as a whole we find striking evidence of downsizing in quasar host galaxy formation: low-luminosity quasars with absolute magnitudes in the range -22 > IAB > -24 have a comoving star formation rate (derived from 100 μm rest-frame luminosities) peaking between redshifts of 1 and 2, while high-luminosity quasars with IAB < -26 have a maximum contribution to the star formation density at z ~ 3. The volume-averaged star formation rate of -22 > IAB > -24 quasars evolves as (1 + z)2.3±0.7 at z < 2, but the evolution at higher luminosities is much faster reaching (1 + z)10±1 at -26 > IAB > -28. We tentatively interpret this as a combination of a declining major merger rate with time and gas consumption reducing fuel for both black hole accretion and star formation. Herschel is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia with important participation from NASA.