We have used the IRAM Plateau de Bure mm interferometer and the UKIRT 1-5 um Imager Spectrometer to test the connection between the major phases of spheroid growth and nuclear accretion by mapping CO emission in nine submm-detected QSOs at z=1.7-2.6 with black hole (BH) masses derived from near-infrared spectroscopy. With a previously published QSO, we present sensitive CO(3-2) observations of 10 submm-detected QSOs selected at the epoch of peak activity in both QSOs and submm galaxies (SMGs). CO is detected in 5/6 very optically luminous (M_B~-28) submm-detected QSOs with BH masses M_BH~10^9-10^10 Msun, confirming the presence of large gas reservoirs of M_gas~3.4x10^10 Msun. However, we find that their BH masses are ~30 times too large and their surface density is ~300 times too small to be related to typical SMGs in an evolutionary sequence. Conversely, we measure weaker CO emission in four fainter (M_B~-25) submm-detected QSOs with properties, BH masses (M_BH~5x10^8 Msun), and surface densities similar to SMGs. These QSOs appear to lie near the local M_BH/M_sph relation, making them plausible `transition objects in the proposed evolutionary sequence linking QSOs to the formation of massive young galaxies and BHs at high-redshift. We show that SMGs have a higher incidence of bimodal CO line profiles than seen in our QSO sample, which we interpret as an effect of their relative inclinations, with the QSOs seen more face-on. Finally, we find that the gas masses of the four fainter submm-detected QSOs imply that their star formation episodes could be sustained for ~10 Myr, and are consistent with representing a phase in the formation of massive galaxies which overlaps a preceding SMG starburst phase, before subsequently evolving into a population of present-day massive ellipticals. [abridged]
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