论文标题
用于泊松方程的变分量子算法
Variational Quantum algorithm for Poisson equation
论文作者
论文摘要
泊松方程在许多科学和工程领域都有广泛的应用。尽管有一些量子算法可以有效地解决泊松方程，但它们通常需要一台易于故障的量子计算机，这超出了当前技术。在本文中，我们提出了一种差异量子算法（VQA）来求解泊松方程，该方程可以在噪声中间尺度量子（NISQ）设备上执行。详细说明，我们首先采用有限的差异方法将泊松方程转换为线性系统。然后，根据线性系统的特殊结构，我们找到了一个明确的张量产品分解，仅$ 2 \ log n+1 $项目，其系数矩阵的特定简单运算符，其中$ n $是系数矩阵的维度。这意味着所提出的VQA仅需要$ O（\ log n）$测量，这大大减少了量子资源。此外，我们执行量子钟测量，以有效地评估简单操作员的期望值。数值实验表明我们的算法可以有效地求解泊松方程。
The Poisson equation has wide applications in many areas of science and engineering. Although there are some quantum algorithms that can efficiently solve the Poisson equation, they generally require a fault-tolerant quantum computer which is beyond the current technology. In this paper, we propose a Variational Quantum Algorithm (VQA) to solve the Poisson equation, which can be executed on Noise Intermediate-Scale Quantum (NISQ) devices. In detail, we first adopt the finite difference method to transform the Poisson equation into a linear system. Then, according to the special structure of the linear system, we find an explicit tensor product decomposition, with only $2\log n+1$ items, of its coefficient matrix under a specific set of simple operators, where $n$ is the dimension of the coefficient matrix. This implies that the proposed VQA only needs $O(\log n)$ measurements, which dramatically reduce quantum resources. Additionally, we perform quantum Bell measurements to efficiently evaluate the expectation values of simple operators. Numerical experiments demonstrate that our algorithm can effectively solve the Poisson equation.
