A Formulation of a Matrix Sparsity Approach for the Quantum Ordered Search Algorithm

TL;DR

This paper introduces a matrix sparsity approach to optimize semidefinite programming for quantum ordered search algorithms, aiming to lower the upper bound of query complexity by reducing computational resources needed.

Contribution

It develops a novel matrix sparsity method to improve SDP computations, enabling more efficient quantum query bound optimization for the ordered search problem.

Findings

Matrix sparsity reduces SDP computation time

Efficient SDP implementation with sparsity improves bounds

Potential to approach the theoretical lower bound

Abstract

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