What's a "photon shift" and what is it doing in LIGO?
What do you mean by entangling it to a quantum computer? How would you go about doing that?
As for gravitational waves, they are not quantum systems. Or, if they are quantum systems, we don't have a good quantum theory for them yet.
Your question seems extremely speculative.
Why are you interested in solving this particular conundrum?
P.S. Welcome to sciforums.