Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries
Abstract
Description
Generalizing earlier work characterizing the quantum query complexity of computing a function of an unknown classical ``black box'' function drawn from some set of such black box functions, we investigate a more general quantum query model in which the goal is to compute functions of N by N ``black box'' unitary matrices drawn from a set of such matrices, a problem with applications to determining properties of quantum physical systems. We characterize the existence of an algorithm for such a query problem, with given error and number of queries, as equivalent to the feasibility of a certain set of semidefinite programming constraints, or equivalently the infeasibility of a dual of these constraints, which we construct. Relaxing the primal constraints to correspond to mere pairwise near-orthogonality of the final states of a quantum computer, conditional on black-box inputs having distinct function values, rather than bounded-error determinability of the function value via a single measurement on the output states, we obtain a relaxed primal program the feasibility of whose dual still implies the nonexistence of a quantum algorithm. We use this to obtain a generalization, to our not-necessarily-commutative setting, of the ``spectral adversary method'' for quantum query lower bounds.
Dagstuhl Seminar Proceedings 06391, "Algorithms and Complexity for Continuous Problems," ed. S. Dahlke, K. Ritter, I. H. Sloan, J. F. Traub (2006), available electronically at http://drops.dagstuhl.de/portals/index.php?semnr=06391
Dagstuhl Seminar Proceedings 06391, "Algorithms and Complexity for Continuous Problems," ed. S. Dahlke, K. Ritter, I. H. Sloan, J. F. Traub (2006), available electronically at http://drops.dagstuhl.de/portals/index.php?semnr=06391