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 $N imes N$ ``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.
Quantum query complexity semidefinite programming
1-25
Regular Paper
Howard
Barnum
Howard Barnum
10.4230/DagSemProc.06391.3
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