Buffer for 5V, 500ps to 10 ns pulses
td;dr Your requirements are unreasonable. Reduce the load capacitance.

Just to establish some ground rules here. What you want needs major amounts of current to get the job done. 

Capacitance voltage charge from a constant current source is expressed as
\begin{equation}
\ Ut = \frac{(I*t)}C
\end{equation}

Given 5V and 50pF, that works out to 500mA  of current just to get the pulse high within 500ps. And that's probably not too useful for you. So to get something a bit more reasonable like 50ps for 10% rise/fall time you're looking at 5 amperes of current to get the job done. 

We're talking about a major high speed driver here. To look at it other way, your 500ps pulses are 2GHz if they were sinewave, which they're not. Some of the modern fast serial drivers can handle the speed but nowhere near the capacitance. 

If you're really got your heart set at building this, you'd essentially need to design a basestation-level microwave amplifier. This is very much specialty equipment, but does not actually cost that much if you go to ebay. Google for microwave power amplifier. Note that you'd need to go up to 20GHz to get that 10% rise and fall time.

Alternatively positive emitter coupled logic might be coaxed to produce required amounts of oomph but designing such a monster is beyond the scope.