Variational quantum algorithm for the Poisson equation
Variational quantum algorithm for the Poisson equation
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DOI：
10.1103/physreva.104.022418
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发表时间：
2021-08-18
影响因子：
2.9
通讯作者：
Wen, Qiao-Yan
中科院分区：
文献类型：
--
作者：
Liu, Hai-Ling;Wu, Yu-Sen;Wen, Qiao-Yan
泊松方程在科学和工程的许多领域都有广泛的应用。尽管有一些量子算法可以有效求解泊松方程，但它们通常需要容错的量子计算机，这超出了当前技术。我们提出了一种变分量子算法（VQA）来求解泊松方程，该算法可以在嘈杂的中等规模量子设备上执行。具体来说，我们首先采用有限差分法将泊松方程转化为线性系统。然后，根据线性系统的特殊结构，我们找到其系数矩阵在一组特定的简单算子下的显式张量积分解，只有(2 log(2) n + 1)项，其中n是系数矩阵的维数。这意味着所提出的 VQA 需要更少的量子测量，从而大大减少了所需的量子资源。此外，我们设计可观测值来有效评估量子计算机上简单算子的期望值。数值实验表明我们的算法可以求解泊松方程。
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. We propose a variational quantum algorithm (VQA) to solve the Poisson equation, which can be executed on noisy intermediate-scale quantum 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(2) 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 needs fewer quantum measurements, which dramatically reduces the required quantum resources. Additionally, we design observables to efficiently evaluate the expectation values of the simple operators on a quantum computer. Numerical experiments demonstrate that our algorithm can solve the Poisson equation.