Different hybrid quantum-classical algorithms have recently been developed as a near-term way to solve linear systems of equations on quantum devices. However, the focus has so far been mostly on the methods, rather than the problems that they need to tackle. In fact, these algorithms have been run on real hardware only for problems in quantum physics, such as Hamiltonians of a few qubits systems. These problems are particularly favorable for quantum hardware, since their matrices are the sum of just a few unitary terms and since only shallow quantum circuits are required to estimate the cost function. However, for many interesting problems in linear algebra, it appears far less trivial to find an efficient decomposition and to trade it off with the depth of the cost quantum circuits. A first simple yet interesting instance to consider are tridiagonal systems of equations. These arise, for instance, in the discretization of one-dimensional finite element analyses. This work presents a method to solve a class of tridiagonal systems of equations with the variational quantum linear solver (VQLS), a recently proposed variational hybrid algorithm for solving linear systems. In particular, we present a new decomposition for this class of matrices based on both Pauli strings and multi--qubit gates, resulting in less terms than those obtained by just using Pauli gates. Based on this decomposition, we discuss the tradeoff between the number of terms and the near-term implementability of the quantum circuits. Furthermore, we present the first simulated and real-hardware results obtained by solving tridiagonal linear systems with VQLS, using the decomposition proposed.

翻译：近年来，多种混合量子-经典算法被提出，作为在量子设备上求解线性方程组的近期可行方案。然而，目前的研究焦点主要集中在方法本身，而非其所需处理的具体问题。事实上，这些算法仅在量子物理领域的问题（如少数量子比特系统的哈密顿量）中于真实硬件上运行过。这类问题对量子硬件尤为有利，因为其矩阵仅由少数幺正项求和构成，且估计代价函数仅需浅层量子电路。然而，对于线性代数中许多有意义的问题，寻找高效分解方案并在其与代价量子电路的深度之间取得平衡显得远非易事。三对角方程组是首个值得考虑的简单而有趣的实例，例如在一维有限元分析的离散化过程中便会出现此类方程组。本研究提出了一种利用变分量子线性求解器（VQLS）——一种近期提出的用于求解线性方程组的变分混合算法——来求解一类三对角方程组的方法。具体而言，我们针对这类矩阵提出了一种基于泡利字符串与多量子比特门的新型分解方案，其所得项数少于仅使用泡利门的分解结果。基于此分解方案，我们探讨了项数与量子电路近期可实现性之间的权衡关系。此外，我们首次通过使用所提出的分解方案，利用VQLS求解三对角线性方程组，给出了仿真与真实硬件的实验结果。