There's something I don't understand here, an H bridge is usually composed of power devices, Vertically diffused MOS or IGBT usually, either way having sub-picosecond accurate signals with them is like trying to aim a nuclear missile at a 1 mm target, shure you can do it, but the explosion will blow up the entire city anyway so what's the point in doing that, you just need to get close by
getting back to our devices a MOS turn on time you read on the datasheet is not an exact figure, it varies with the PCB layout (gate trace inductance limit the current rise and thus the speed at which your device switched on/off), mosfet capacitances (in turn influenced by the operating voltages, process and mos geometry variation at the foundry) and so on, the worst thing is that for all these values you get most of the times only typical figures on the datasheet no min/max or graph.
and while some of this variations are static and can be calibrated out some others aren't (like the various capacitances vs VDS), and even if the dependency isn't that strong even a 0.0001 change in turn on time will have on your load by far much greater effects than what your FPGA/MCU will have
and I dont get the energy bit either, jitter is random in nature (assuming independent, gaussian distribution jitter on turn on and turn off of the H-bridge, for lack of better data) you have 0.5 probability that the jitter will be in the same direction at the two ends, having no net effect on the time the load is energized, and 0.25 prob that the jitter will actually decrease the energy you use on the load (by turning on later and turning off before) and only 0.25 prob that the energy will go up, so i think designing around that is a waste of time