It troubles me that everyone went to so much trouble to point out it's a terrible idea, but no one actually answered the question...
First off, is there a way to accomplish supersonic arrows in a non-supernatural way.. no.. but for different reasons than you think. Could we build a mechanical assist longbow? Sure. But when you think about what a bowstring moving at supersonic speeds would do to your face and arm, shiver
For this to even be feasible in reality, you'd need to invent some sort of magnetic launch mechanism because the string would mess you up pretty badly.
But in a supernatural setting?
First understand that strength adjusted bows have existed forever in Dungeons and Dragons. Second, to answer your question accurately would require some specifications..
For one.. You'd need a special bow, probably notched, so arrows can fly with minimal flexing.. it would also need to be enchanted to protect you from the bowstring and sonic crack in your face.
Second.. you need to decide what look you are going for.. are you shooting for the long long range sniper? Because now you have a range of about 6,000 feet.. or are you going for the barbarian who can shoot down castles and blast people out of a forest?
An arrow would need to be a mythological metal to survive.. so either something like a 20,000 grain (almost 3lbs) adamantine arrow.. or at the truly ridiculous end, a 27lbs ballista bolt. I'm going to assume you're going for the more "normal" adamantine arrows (which would be ridiculously expensive but almost invincible!)
Assuming a lot, I'd guess a heavy arrow (600 grain) made fully of adamantite would be about 15000 grain (about 2.25lbs, HEAVY! Note that adamantite, steel and dense ironwood have about the same density). Assuming a 75cm (29 inch) draw and an acceleration TO the speed of sound (thus minimum power) giving about 106,000 m/s/s of acceleration. Force equals mass (0.972kg) times acceleration (106666 m/s/s) gives 1,036,793 Newtons of force, or a draw strength of about 232,000 lbs, or a minimum strength of 7,000 in D&D 5e.
But fret not! There's magic afoot!
If you use a bow of FORCE then you would only need a STR score of about 21 to fire an arrow (assuming the arrow is normal at draw and until after release) at supersonic speeds and 6k range. Now THAT would be something to see no?
I hope that answered the question.
And now I really want a silenced sniper's bow of supersonic force.