Optimization and noise analysis of the quantum algorithm for solving one-dimensional Poisson equation

Solving differential equations is one of the most promising applications of quantum computing. Recently we proposed an efficient quantum algorithm for solving one-dimensional Poisson equation avoiding the need to perform quantum arithmetic or Hamiltonian simulation. In this paper, we further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. To this end, we first optimize the quantum 1D-Poisson solver by developing a new way of performing the sine transformation. The circuit depth for implementing the sine transform is reduced from n2 to n without increasing the total qubit cost of the whole circuit, which is achieved by neatly reusing the additional ancillary quits. Then, we analyse the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit. We find that the phase damping noise has little effect on our algorithm, while the bit flip noise has the greatest impact. In addition, threshold errors of the quantum gates are obtained to make the fidelity of the circuit output being greater than 90%. The results of noise analysis will provide a good guidance for the subsequent work of error mitigation and error correction for our algorithm. The noise-analysis method developed in this work can be used for other algorithms to be executed on the NISQ devices.