LED driver with very fast start up time

Dear Sir or Madam,
we are looking for a LED Boost driver with fast start time.
It's difficult to search for this parameter.
This are our specs:
- USB input, 4,5 - 5,5V
- LED voltage 10V - 15V, or two channels 5 - 7V
- LED current 2A pulse
- clean rectangular pulse with 4ms pulsewith
=> problem: Start up behavior in the range of <=10 µs

At the input we will place storage capacitors to not overload usb.

Please can you help us finding the right part? What start up conditions are possible?
What efficiency is possible? 95%?


Regards
Martin Hermann