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The solution of linear systems of equations is a very frequent operation andthus important in many fields. The complexity using classical methods increaseslinearly with the size of equations. The HHL algorithm proposed by Harrow etal. achieves exponential acceleration compared with the best classicalalgorithm. However, it has a relatively high demand for qubit resources and thesolution $\left| x \right\rangle $ is in a normalized form. Assuming that theeigenvalues of the coefficient matrix of the linear systems of equations can berepresented perfectly by finite binary number strings, three hybrid iterativephase estimation algorithms (HIPEA) are designed based on the iterative phaseestimation algorithm in this paper. The complexity is transferred to themeasurement operation in an iterative way, and thus the demand of qubitresources is reduced in our hybrid algorithms. Moreover, the solution is storedin a classical register instead of a quantum register, so the exactunnormalized solution can be obtained. The required qubit resources in thethree HIPEA algorithms are different. HIPEA-1 only needs one single ancillaryqubit. The number of ancillary qubits in HIPEA-2 is equal to the number ofnondegenerate eigenvalues of the coefficient matrix of linear systems ofequations. HIPEA-3 is designed with a flexible number of ancillary qubits. TheHIPEA algorithms proposed in this paper broadens the application range ofquantum computation in solving linear systems of equations by avoiding theproblem that quantum programs may not be used to solve linear systems ofequations due to the lack of qubit resources.

Hybrid algorithms to solve linear systems of equations with limited qubit resources