Abstract
The Poisson equation has many applications across the broad areas of science and engineering.
Most quantum algorithms for the Poisson solver presented so far either suffer from lack of accuracy
and/or are limited to very small sizes of the problem, and thus have no practical usage. In this regard, our previous work showed
a proof-of-concept demonstration in advancing quantum Poisson solver algorithm and validated preliminary results for a simple case of 3 × 3 problem.
In this work, we delve into comprehensive research details, presenting the results on up to 15 × 15 problems that include step-by-step improvements
in Poisson equation solutions, scaling performance, and experimental exploration. In particular, we demonstrate the implementation of eigenvalue
amplification by a factor of up to 2 8 , achieving a significant improvement in the accuracy of our quantum Poisson solver and comparing that to the exact solution.
Additionally, we present success probability results, highlighting the reliability of our quantum Poisson solver.
Moreover, we explore the scaling performance of our algorithm against the circuit depth and width, demonstrating how our approach scales
with larger problem sizes, and thus further solidifies the practicality of easy adaptation of this algorithm in real-world applications.
We also discuss a multi-level strategy for how this algorithm might be further improved to explore much larger problems with greater performance.
Finally, through our experiments on the IBM quantum hardware, we conclude that though overall results on the existing NISQ hardware are dominated
by the error in the CNOT gates, this work opens a path to realizing a multidimensional Poisson solver on near-term quantum hardware.
