Implementing and Evaluating the Efficiency of a Quantum Linear Systems Algorithm

Osama M. Raisuddin, Suvranu De

Dhruv Sood, Nilmani Mathur, Vikram Tripathi

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.

Adrian Harkness, Kate Saltovets, Mohammadhossein Mohammadisiahroudi +1 more

Dhruv Sood, Nilmani Mathur A, Vikram Tripathi

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.

Simran Jakhodia, Babita Jajodia

Researchers are currently working on computational solutions based on quantum systems to accelerate the speed of complex mathematical models. This work presented how to formulate complex computational problems as a quantum system of linear equations and find solutions using Quantum Linear System Algorithm (QLSA), also called Quantum Harrow-Hassidim-Lloyd (HHL) algorithm. This paper showed experimental evaluation of multiple problem statements (curve-fitting functions, interpolating polynomials) as a quantum system of linear equations that involve computation of Vandermonde matrices as co-efficient matrices on IBM Quantum Information Software Kit for Quantum Computation (QISKit) platform. Along with a few examples demonstrating its evaluation on diagonal, Hermitian, and Non-Hermitian matrices as co-efficient matrices. The fidelity is used as a measure of performance for comparing the accuracy of quantum results with respect to existing classical solutions on IBM QISKit and drawing conclusions from the experimental results. Experimental evaluation shows that the fidelity depends on the sparsity of the input matrices and therefore the results vary depending on those matrices.

Osama M. Raisuddin, Suvranu De