Semidefinite programming characterization and spectral adversary method for quantum complexity with noncommuting unitary queries

Abstract

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 $N 	imes N$ ``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 query and error, 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 the various black-box inputs, rather than bounded-error distinguishability,
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.