I think this is a tough one.
Imagine a rocket is produced which is propelled by a matter-anti-matter device. Its sole purpose is to reach high velocity. Which means the mass can be kept low, say $m$ kilograms. Let's assume it starts in an ideal, infinite, and flat spacetime. By ideal, I mean that there is no real matter, dark matter, or dark energy present.
The rocket's thrust is constant, so when the rocket approaches relativistic speed, the acceleration gets less. Let's assume further that the rocket's restmass stays constant. The mass of the matter and anti-matter (for example, electrons and positrons) is insignificant in comparison to the total restmass of the rocket. The initial velocity is zero and the initial acceleration $a$.
My question is simple: How much (kilogram) anti-matter is needed to reach a velocity of $P$% of the speed of light?
In other words, what does a general formula to calculate this for every end speed look like?
P.S I assume constant thrust (backward force) contrary to a constant acceleration. The answer of course has to be the same but it takes longer to reach the speed required.