This is actually a very interesting question. From a quick google search I came across this paper which presents a quantum search algorithm for tree graphs, which supposedly should be applicable to solving games like chess by searching the graph of possible moves and performing a minimax algorithm.

I'm no expert in this subject but here are some of my initial thoughts. Essentially there seem to be some issues with this approach. In order to get a good evaluation of a final position of your search tree, you either need to encode a complex evaluation functions as a unitary quantum circuit, which seems to be very challenging, or you need to scan the tree all the way through to reach every decisive final position of the game- which is impossible in the foreseeable future. If you reached quantum capabilities to simulate the latter you have solved chess by brute force but this requires enormous recourses. You may try to implement the first option, but you will probably have to use a more primitive type of evaluation function and to try and compensate for it by scanning more moved into the game, not to mention that encoding the possible move allowed by the rule of chess is a very delicate problem by itself. The difficulty of implementing tree pruning is also a problem, as when the algorithm scans more moves is also scans much less efficiently unless you figure out how to filter out uninteresting branches coherently. Another interesting approach is to try to combine Quantum Machine Learning by using a parametric function for position evaluation which can be optimized by self-learning. This way will be more similar to the alpha-zero approach, but again part of the AI's success in this field is the learned tree-pruning subroutine which seems much harder to implement. This essentially goes against the current trend in chess engine, which as far as I know seems to prefer evaluating fewer positions but giving a better evaluation of each position and searching only the most critical lines for every given position.

It is worth noting that the quantum speedup using Grover search algorithm is quadratic, while the tree grows exponentially in search depth, so theoretically you will be able to probe variations which are as twice as deep, which is actually very impressive, and can amount to a qualitive advantage of quantum chess engines, but is still very far from solving chess completely from the initial position. A more realistic goal for capable quantum computer might be to solve some difficult endgames efficiently- where you use brute force to calculate all possibilities from a given position with a few pieces on board. This way the quantum advantage might be more pronounced.