Quantum fast Poisson solver: the algorithm and complete and modular circuit design

Abstract

The Poisson equation has applications across many areas of physics and engineering, such as the dynamic process simulation of ocean current. Here we present a quantum Fast Poisson Solver, including the algorithm and the complete and modular circuit design. The algorithm takes the HHL algorithm as the template. The controlled rotation is performed based on the arc cotangent function which is evaluated by the Plouffe's binary expansion method. And the same method is used to compute the cosine function for the eigenvalue approximation in phase estimation. Quantum algorithms for solving square root and reciprocal functions are developed based on the non-restoring digitrecurrence method. These advances make the algorithm's complexity lower and the circuit-design more modular. The number of the qubits and operations used by the circuit are O(dlog 2 (ε -1 )) and O(dlog 3 (ε -1 )), respectively. We demonstrate our circuits on a quantum virtual computing system installed on the Sunway TaihuLight supercomputer. This is an important step toward practical applications of quantum Fast Poisson Solver in the near-term hybrid classical/quantum devices.

A quantum circuit simulator and its applications on Sunway TaihuLight supercomputer

“…Solving the Poisson equation, thus, constitutes the most computationally intensive part of the ocean current simulation. We develop a quantum algorithm for solving the multi-dimensional Poisson equation 27 . It could provide an exponential speedup to some degree over the classical counterparts.…”

“…To demonstrate the correctness of the algorithm, we propose a simplified version of the circuit with four discretized points 27 . The circuit consists of 38 qubits and 800 gates.…”

“…It consists of three main stages, i.e., phase estimation, controlled rotation and uncomputation. The complexity of our algorithm is O d log 2 (ε −α ) in qubits and O κd log 3 (ε −α ) in quantum operations, where ε is the error To demonstrate the correctness of the algorithm, we propose a simplified version of the circuit with four discretized points 27 . The circuit consists of 38 qubits and 800 gates.…”

Detailed Account of Complexity for Implementation of Circuit-Based Quantum Algorithms

“…There are many quantum algorithms whose output state has coherence in the computational basis. There are algorithms to solve partial differential equations [53][54][55][56][57][58][59], linear differential equations [12][13][14], nonlinear differential equations [60], linear system of equations (also named quantum linear problem) [61,62]. In these examples, QST may be required depending on the level of detail expected to be known.…”

Quantum algorithm for the advection–diffusion equation simulated with the lattice Boltzmann method

“…In case of inhomogeneous linear differential equations, Berry [26] and Childs et al [27] first formed a system of linear equations by discretizing the differential equation using the finite difference method (FDM), which then are solved by applying the quantum algorithm to the linear systems of equations. Examples of this approach for the Vlasov equation are presented by Costa et al [28], while in case of Poisson equation Cao et al [29] and Wang et al [30] applied HHL [5] algorithm for solving system of linear equations. Another approach, where the Taylor expansion of the analytical solution of the linear differential equations (LDE) is described by the quantum states and the corresponding operators, is firstly proposed by the Berry et al [31], while application on 4-dimensional LDE with a 4×4 non-unitary matrix is done by Xin et al [32].…”