You're going to find a lot of physics gets in the way.
The most important detail will be that arrows bend while being fired. Because all of the propulsion is coming from the back of the arrow, the arrow is naturally unstable during firing. This is something archers are aware of at normal archery speeds. However, you will need to break the sound barrier while the arrow is still under power, so you'll need a lot of power in this unstable time.
Let's pretend your arrow is made of unobtanium, and is rigid enough. Let's also pretend your bow has a good enough "dry shot" velocity to accelerate anything to those speeds (which I think may be reasonable, with enough clever geometry). A reasonable mass arrow is around 200 grains (13 grams). In the interest of approximations, let's round down to 10grams, or 0.01kg to make the math easier. The draw length of an average bow might be around 0.75m. We can do the calculations to see how much energy an arrow has to have at the sound barrier: 340 m/s, and then use that to figure out how much force is needed to do that work over the draw length. E=1/2mv^2 = 578J. F=W/d = 770N. That's roughly 175 pounds of draw strength, to accelerate the arrow with no air friction or anything.
That's about triple the strength of a human right now. However it fails to take into account a lot of things, least of all being the inefficiency of the bow near the speed of sound (you're basically going to have to make the string into a bull whip).
However, that's just to get to the speed of sound. The drag of a projectile at supersonic speeds is quite high, about three times higher than you'd expect for a well designed projectile. Consider that a 9mm bullet is supersonic when it comes out of the barrel, but rapidly slows to subsonic. You would actually have to accelerate the arrow to a much higher speed to remain supersonic in flight. Trying to fire at mach 2 would require 4x the force.
All of that presumes access to an unobtanium shaft that can take the stress of launching, a carefully constructed bow with pulls quite stronger than any human can handle, and probably ignores a half a dozen limitations that appear in the transonic region that I am simply not accounting for.
The arrow shafts seem like the most difficult part to solve. Modern archers pay top dollar for ultra-exotic composite arrow shafts to minimize flex at human speeds. The material requirements for that arrow shaft supersonic may simply be outside the realm of known materials.
Want to shoot an arrow supersonic without these problems? Try a sabot. You can use them to basically put a sharp small object (like an arrow) into a gun. The sabot holds the shape as the explosives project the entire construct out of the barrel. Then the sabot falls away and lets the sharp thing fly.