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The Poisson equation has many applications across the broad areas of scienceand engineering. Most quantum algorithms for the Poisson solver presented sofar either suffer from lack of accuracy and/or are limited to very small sizesof the problem, and thus have no practical usage. Here we present an advancedquantum algorithm for solving the Poisson equation with high accuracy anddynamically tunable problem size. After converting the Poisson equation to alinear system through the finite difference method, we adopt the HHL algorithmas the basic framework. Particularly, in this work we present an advancedcircuit that ensures the accuracy of the solution by implementing non-truncatedeigenvalues through eigenvalue amplification, as well as by increasing theaccuracy of the controlled rotation angular coefficients, which are thecritical factors in the HHL algorithm. Consequently, we are able to drasticallyreduce the relative error in the solution while achieving higher successprobability as the amplification level is increased. We show that our algorithmnot only increases the accuracy of the solutions but also composes morepractical and scalable circuits by dynamically controlling problem size in NISQdevices. We present both simulated and experimental results and discuss thesources of errors. Finally, we conclude that though overall results on theexisting NISQ hardware are dominated by the error in the CNOT gates, this workopens a path to realizing a multidimensional Poisson solver on near-termquantum hardware.

Advancing Algorithm to Scale and Accurately Solve Quantum Poisson Equation on Near-term Quantum Hardware