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Quantum algorithms for solving linear systems of equations have generatedexcitement because of the potential speed-ups involved and the importance ofsolving linear equations in many applications. However, applying thesealgorithms can be challenging. The Harrow-Hassidim-Lloyd algorithm andimprovements thereof require complex subroutines suitable for fault-toleranthardware such as Hamiltonian simulation, making it ill-suited to currenthardware. Variational algorithms, on the other hand, involve expensiveoptimization loops, which can be prone to barren plateaus and local optima. Wedescribe a quantum algorithm for solving linear systems of equations thatavoids these problems. Our algorithm is based on the Woodbury identity, whichanalytically describes the inverse of a matrix that is a low-rank modificationof another (easily-invertible) matrix. This approach only utilizes basicquantum subroutines like the Hadamard test or the swap test, so it iswell-suited to current hardware. There is no optimization loop, so barrenplateaus and local optima do not present a problem. The low-rank aspect of theidentity enables us to efficiently transfer information to and from the quantumcomputer. This approach can produce accurate results on current hardware. Asevidence of this, we estimate an inner product involving the solution of asystem of more than 16 million equations with 2% error using IBM's Aucklandquantum computer. To our knowledge, no system of equations this large haspreviously been solved to this level of accuracy on a quantum computer.

A near-term quantum algorithm for solving linear systems of equations based on the Woodbury identity