As an initial matter, let's ask "what is the classical computational complexity of solving 'mate-in-$n$' type games?" For example, is it even in $\mathcal{NP}$ to know, given a certain chess position, that white can mate in $10$ or fewer moves?
It's been known for a while that we can consider such questions as a "quantified boolean formula" (QBF) question. Let's rephrase our question as:
$$\exists w_1\forall b_1\exists w_2\forall b_2\cdots\exists w_n:\phi(w_1,b_1,w_2,b_2,\cdots,w_n)?$$
This "mate-in-$n$" statement can be read as "given a state of the board $\phi$ encoding the rules of chess, does there exist a move by white such that for moves by black, there is a countermove by white such that... the moves applied to the board $\phi$ will lead to a mate by white?"
This "mate-in-$n$" is precisely a way to think about the polynomial hierarchy $\mathcal{PH}$. The mate-in-$10$ is at the $10^{th}$ level of the hierarchy (or maybe it's the $20^{th}$) because there are $20$ iterations of for-alls and there-exists. $\phi$ is easy to evaluate (polynomial evaluation of whether there's a mate or not). The number of qubits would be polynomial in $n$ I think (high thousands?).
Letting $n$ vary polynomialy distinguishes $\mathcal{PH}$ from $\mathcal{PSPACE}$. Magically, although we know that $\mathcal{BQP}\subseteq\mathcal{PSPACE}$, it's likely that $\mathcal{BQP}$ and $\mathcal{PH}$ are incomparable. I think the implication is that certain well-framed QBF-style questions can be answered faster than classical algorithms.
For chess proper, most games can be completed in about $50$ moves or less. So we can say "is there a mate-in-$50$ for white, given the starting position?"
Chess has a lot of asymmetry that may make it difficult to frame in the right way. Weichi (Go) may be a better candidate for pondering. I suspect a toy game, based on forrelation, can be built where a quantum computer can outperform a classical computer.
EDIT
To think about how to utilize such an algorithm, let's take the QBF a little bit more. The output of the QBF is either a "YES" (a forced mate is possible) or "NO" (a forced mate is not possible).
Say we are white, and we have a quantum QBF-solver that can be fed a given board position with black to move, and will spit out whether there's a winning strategy for black (i.e. YES or NO). It is our turn to move.
We can cycle through all of the available moves for white by making a putative move, and ask our quantum QBF-solver whether black has a winning response. If we find a move where black doesn't have a winning response we can take that line.
In the comments, you suggest that chess isn't likely played out in $50$ moves starting from the initial board, and that $6000$ moves is more likely.
Nonetheless even accepting $6000$ is the better estimate, I maintain that the number of qubits is polynomial in the depth of the tree you are willing to review, otherwise you haven't got a practical algorithm.