NAND-Trees, Average Choice Complexity, and Effective Resistance

We show that the quantum query complexity of evaluating NAND-tree instances with average choice complexity at most $W$ is $O(W)$, where average choice complexity is a measure of the difficulty of winning the associated two-player game. This generalizes a superpolynomial speedup over classical query complexity due to Zhan et al. [Zhan et al., ITCS 2012, 249-265]. We further show that the player with a winning strategy for the two-player game associated with the NAND-tree can win the game with an expected $\widetilde{O}(N^{1/4}\sqrt{{\cal C}(x)})$ quantum queries against a random opponent, where ${\cal C }(x)$ is the average choice complexity of the instance. This gives an improvement over the query complexity of the naive strategy, which costs $\widetilde{O}(\sqrt{N})$ queries. The results rely on a connection between NAND-tree evaluation and $st$-connectivity problems on certain graphs, and span programs for $st$-connectivity problems. Our results follow from relating average choice complexity to the effective resistance of these graphs, which itself corresponds to the span program witness size.