A B757 First Officer reported auto pilot and hydraulic power failure on initial climb and return to departure airport.

Narrative

Normal push; start; taxi; and takeoff. When the gear handle was raised after takeoff (or shortly thereafter); upper EICAS display went blank; the right electric hydraulic pump press light illuminated with associated EICAS message on lower screen; the left center fuel pump press light illuminated; and the auto throttle disconnected with subsequent cautions. We continued the climb out; cleaned up the aircraft; and ran the After-Takeoff Checklist. As we climbed and cleaned up the aircraft; we noticed there was no EPR (Engine Pressure Ratio) displayed for the left engine. We discussed the status of the left engine; turned on the standby engine instruments (which also had no readout for ERP on the left engine); and determined the left engine was operating normally. We analyzed the issues and discussed the possibility of an ac bus failure or other electrical failure; but the left ac bus and generator were displaying normally. We leveled at 10;000 MSL and ran the HYD ELEC PUMP L (R) QRH. There was no change in the situation upon checklist completion. During this time; we also noticed and discussed the autopilot system making jerky and frequent aileron inputs; and we had an overall lack of confidence in the autopilot based on control wheel feedback and the autopilot inputs. We decided to hand fly for the remainder of the sortie. We elected to return to ZZZ; [requested priority handling]; and asked for delayed vectors to [Runway] XXR. This is also when we discovered the center fuel tank was not feeding both engines correctly as a 1000lb fuel imbalance had developed. We had 11-12000 lbs of fuel in the center tank. We turned off both center fuel pumps and cross fed from the right fuel tank to correct the imbalance. The Captain transferred aircraft control to me and contacted Dispatch. The Captain conferred with Maintenance Control and Dispatch who agreed a return to ZZZ was the best course of action. The overweight landing was also discussed with Maintenance Control and Dispatch as well as between the two of us. The Captain set up for the approach and briefed the ILS XXR approach. The aircraft controls were transferred back to the Captain; we accomplished a descent checklist; and the approach and landing were uneventful (it was a very smooth touchdown). We cleared the runway and taxied back to the gate uneventfully.