the equations u need:
(1) time constant of load (τ) =Rload*Cload=50*5=250 ps
(2) for any N-bit resolution converter, the following equation must be hold.
2^N*LSB*exp(-t/τ) < LSB/2
==> t > τ*(N+1)*ln(2)
==> 1/f > τ*(N+1)*ln(2)
(3) in ur case, f=1GHz, τ=250ps, ln(2)~=0.7, ==> N < 4.8
So, in theorem, u will never achieve 1GHz operation with 7 bit resolution
(4) to increase operation, u should decrease τ first, either Rload or Cload should
be decreased, after that Rload is calculated, then from your full scale (FS) spec. u can calculate the required current = FS/Rload