The Harrow-Hassidim-Lloyd (HHL) algorithm is a quantum algorithm for solving systems of
linear equations that, in principle, offers an exponential improvement in scaling with the system
size compared to classical approaches. In this work, we investigate the practical implementation
and optimisation of the HHL algorithm with a focus on improving its performance on near-
term quantum simulators. After outlining the algorithm, we examine two optimisation strategies
aimed at improving fidelity and scalability: Suzuki-Trotter decomposition of the Hamiltonian
evolution operator and a block-encoding approach that embeds the problem matrix into a larger
unitary operator. The performance of these methods is evaluated through simulations on matrices
with varying sparsity, including diagonal, tridiagonal, moderately dense, and fully dense cases.
Our results show that while HHL achieves near-ideal fidelity for highly structured matrices,
performance degrades as sparsity decreases due to the increasing cost of Hamiltonian simulation
and reduced post-selection probability due to higher condition number. Block encoding is found
to provide improved fidelity for moderately dense matrices, whereas Trotterisation offers a qubit-
efficient approach for sparse systems. These results highlight the importance of matrix structure
in determining the practical efficiency of HHL and inform future implementations that combine
algorithmic optimisation with hardware-aware design.