We develop a novel approach for efficiently applying variational quantum linear solver (VQLS) in context of structured sparse matrices. Such matrices frequently arise during numerical solution of partial differential equations which are ubiquitous in science and engineering. Conventionally, Pauli basis is used for linear combination of unitary (LCU) decomposition of the underlying matrix to facilitate the evaluation the global/local VQLS cost functions. However, Pauli basis in worst case can result in number of LCU terms that scale quadratically with respect to the matrix size. We show that by using an alternate basis one can better exploit the sparsity and underlying structure of matrix leading to number of tensor product terms which scale only logarithmically with respect to the matrix size. Given this new basis is comprised of non-unitary operators, we employ the concept of unitary completion to design efficient quantum circuits for computing the global/local VQLS cost functions. We compare our approach with other related concepts in the literature including unitary dilation and measurement in Bell basis, and discuss its pros/cons while using VQLS applied to Heat equation as an example.

翻译：我们提出了一种新方法，用于在结构化稀疏矩阵场景下高效应用变分量子线性求解器（VQLS）。这类矩阵在求解科学与工程中普遍存在的偏微分方程时频繁出现。传统上，泡利基被用于对底层矩阵进行线性组合中的酉算子分解（LCU），以促进全局/局部VQLS代价函数的评估。然而，泡利基在最坏情况下可能导致LCU项的数量随矩阵规模呈二次增长。我们证明，通过使用替代基可以更好地利用矩阵的稀疏性和底层结构，使得张量积项的数量仅随矩阵规模呈对数增长。由于该新基由非酉算子组成，我们引入酉补全的概念来设计计算全局/局部VQLS代价函数的高效量子电路。我们将该方法与文献中其他相关概念（包括贝尔基下的酉膨胀与测量）进行对比，并以热方程为例讨论了其在应用VQLS时的优缺点。