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Solving linear systems is of great importance in numerous fields. Inparticular, circulant systems are especially valuable for efficiently findingnumerical solutions to physics-related differential equations. Current quantumalgorithms like HHL or variational methods are either resource-intensive or mayfail to find a solution. We present an efficient algorithm based on convexoptimization of combinations of quantum states to solve for banded circulantlinear systems whose non-zero terms are within distance $K$ of the maindiagonal. By decomposing banded circulant matrices into cyclic permutations,our approach produces approximate solutions to such systems with a combinationof quantum states linear to $K$, significantly improving over previousconvergence guarantees, which require quantum states exponential to $K$. Wepropose a hybrid quantum-classical algorithm using the Hadamard test and thequantum Fourier transform as subroutines and show its PromiseBQP-hardness.Additionally, we introduce a quantum-inspired algorithm with similarperformance given sample and query access. We validate our methods withclassical simulations and actual IBM quantum computer implementation,showcasing their applicability for solving physical problems such as heattransfer.

Hybrid quantum-classical and quantum-inspired classical algorithms for solving banded circulant linear systems