  One specific subset of quantum algorithms is Grovers Ordered Search Problem
(OSP), the quantum counterpart of the classical binary search algorithm, which
utilizes oracle functions to produce a specified value within an ordered
database. Classically, the optimal algorithm is known to have a $\log_2 N$
complexity; however, Grovers algorithm has been found to have an optimal
complexity between the lower bound of $((\ln N-1)/\pi \approx 0.221\log_2 N)$
and the upper bound of $0 .433\log_2 N$. We sought to lower the known upper
bound of the OSP. With [E. Farhi et al, arXiv:quant-ph/9901059], we see that
the OSP can be resolved into a translational invariant algorithm to create
quantum query algorithm restraints. With these restraints, one can find Laurent
polynomials for various $k$ -- queries -- and $N$ -- database sizes -- thus
finding larger recursive sets to solve the OSP and effectively reducing the
upper bound. These polynomials are found to be convex functions, allowing one
to make use of convex optimization to find an improvement on the known bounds.
According to [A. Childs et al, arXiv:quant-ph/0608161v1], semidefinite
programming, a subset of convex optimization, can solve the particular problem
represented by the constraints . We were able to implement a program abiding to
their formulation of a semidefinite program (SDP), leading us to find that it
takes an immense amount of storage and time to compute. To combat this setback,
we then formulated an approach to improve results of the SDP using matrix
sparsity. Through the development of this approach, along with an
implementation of a rudimentary solver, we demonstrate how matrix sparsity
reduces the amount of time and storage required to compute the SDP -- overall
ensuring further improvements will likely be made to reach the theorized lower
bound
