Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries
<|reference_start|>Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries: 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.<|reference_end|>
@article{barnum2007semidefinite,
    title={Semidefinite programming characterization and spectral adversary method
  for quantum complexity with noncommuting unitary queries},
    author={Howard N. Barnum},
    journal={arXiv preprint arXiv:quant-ph/0703141},
    year={2007},
    archivePrefix={arXiv},
    eprint={quant-ph/0703141},
    primaryClass={quant-ph cs.CC}
}