It's quite inconvenient to drive the gates of MOSFETs to improve the efficiency of a full-wave rectifier. Here's a paper where the authors have investigated this: "A novel high efficient approach to input bridges".
The losses at 2A are not all that great (a couple percent), especially if you use Schottky diodes, so it's probably not worthwhile to attempt this at present.
You might find "synchronous" a useful search term in your further research. That's normally what active power rectifiers are called, whereas "ideal rectifier" normally is in reference to signal processing absolute value circuits.