Solving linear systems is of great importance in numerous fields. In particular, circulant systems are especially valuable for efficiently finding numerical solutions to physics-related differential equations. Current quantum algorithms like HHL or variational methods are either resource-intensive or may fail to find a solution. We present an efficient algorithm based on convex optimization of combinations of quantum states to solve for banded circulant linear systems whose non-zero terms are within distance $K$ of the main diagonal. By decomposing banded circulant matrices into cyclic permutations, our approach produces approximate solutions to such systems with a combination of quantum states linear to $K$, significantly improving over previous convergence guarantees, which require quantum states exponential to $K$. We propose a hybrid quantum-classical algorithm using the Hadamard test and the quantum Fourier transform as subroutines and show its PromiseBQP-hardness. Additionally, we introduce a quantum-inspired algorithm with similar performance given sample and query access. We validate our methods with classical simulations and actual IBM quantum computer implementation, showcasing their applicability for solving physical problems such as heat transfer.

翻译：求解线性系统在众多领域具有重要价值。其中，循环系统在高效求解物理相关微分方程的数值解方面尤为关键。当前诸如HHL或变分法等量子算法要么资源消耗巨大，要么可能无法获得解。本文提出一种基于量子态组合凸优化的高效算法，用于求解非零项位于主对角线距离$K$范围内的带状循环线性系统。通过将带状循环矩阵分解为循环置换，我们的方法能够以与$K$呈线性关系的量子态组合逼近此类系统的解，相较于此前需要与$K$呈指数关系的量子态的收敛性保证，实现了显著改进。我们设计了一种基于Hadamard测试和量子傅里叶变换子程序的混合量子-经典算法，并证明其属PromiseBQP-hard复杂度类。此外，我们引入一种在给定样本与查询访问条件下具有相似性能的量子启发式经典算法。通过经典仿真及在IBM量子计算机上的实际部署验证了方法的有效性，展示了其在求解热传导等物理问题中的适用性。