An Empirical Analysis on the Effectiveness of the Variational Quantum Linear Solver

Variational Quantum Algorithms (VQAs) have emerged as promising methods for tackling complex problems on near-term quantum devices. Among these algorithms, the Variational Quantum Linear Solver (VQLS) addresses linear systems of the form $Ax=b$, aiming to prepare a quantum state $|x\rangle$ such that $A|x\rangle$ is proportional to the quantum state corresponding to $b$. A key advantage of VQLS is its use of amplitude encoding, which requires a number of qubits that scales logarithmically with the linear system size. However, the existing literature has primarily focused on linear systems of limited size or with a specific structure. In this study, we extend the application of VQLS to more general and larger problem instances, including problems where state preparation is non-trivial and problems within the real domain of fluid dynamics. Our investigation reveals some critical challenges inherent to VQLS, including the need for a sufficiently expressive ansatz, the large number of circuit executions required to estimate the cost function, and the high gate count in the circuits in the most general setting. Our analysis highlights the obstacles that need to be addressed for a broader application of VQLS and concludes that further research is necessary to fully leverage the algorithm's capabilities in addressing real-world problems.