Classical combinations of quantum states for solving banded circulant linear systems

Solving linear systems is of great importance in numerous fields. Proposed quantum algorithms for preparing solutions for linear systems include the HHL algorithm with subsequent refinements and variational methods. Circulant linear systems appear in many physics-related differential equations. An interesting case is banded circulant linear systems whose non-zero terms are within distance K of the main diagonal. For these systems, we propose an approach based on the classical combination of quantum states (CQS) method relying on convex optimization against the available analytical solution. From decompositions into cyclic permutations, the solution can be approximately represented by a classical combination of a polynomial number of quantum states. We validate our methods using classical simulations as well as execution on an IBM quantum computer. While in the setting of this paper, efficient classical algorithms are available, our results demonstrate the potential applicability of the CQS method for solving physics problems such as heat transfer.