We show a connection between NAND-tree evaluation and st-connectivity problems on certain graphs to generalize a superpolynomial quantum speedup of Zhan et al. for a promise version of NAND-tree formula evaluation. In particular, 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. Our results follow from relating average choice complexity to the effective resistance of these graphs, which itself corresponds to the span program witness size. These connections suggest an interesting relationship between st-connectivity problems and span program algorithms, that we hope may have further applications.

This is joint work with Shelby Kimmel.