Blake Holman

We introduce the parallel reversible pebbling game on directed graphs for constructing parallel quantum circuits that are efficient with respect to amortized space-time complexity (equivalently named cumulative complexity (CC)), a stronger metric than the conventional space-time complexity used for parallel algorithms. Our main result is a mapping from irreversible algorithms for computing a function, to quantum algorithms for computing the function in superposition, with just a sub-polynomial overhead in cumulative complexity. Thus, to construct a CC-efficient quantum oracle for a function, it suffices to solve the simpler problem of designing a CC-efficient classical algorithm for the function. This transformation also allows us to leverage the vast body of work on classical pebbling games for developing parallel quantum circuits with low amortized space-time complexity, given the data-dependency graph of the problem.