Nothing. Sure, they'll solve chess in a hundred years or so, but the number of possible variations will simply be beyond human capacity to memorize. We already can't beat the top engines. Oh I suppose super GMs might try to take advantage of some "winning" lines, but I doubt they'll succeed. And I suppose book openings will extend into near infinity, but mere mortals won't be able to take advantage. My $.02. 
Quantum computing and chess
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And then in whatever form public discussions will take place in 100 years, we'll have posts from kids complaining "I have the Quantum Chess Encyclopedia and it states that c5 is a guaranteed loss for black against e4 but a friend of mine keeps beating me with it. What can I do?" LOL
What will quantum computing mean for chess?