  The solution of linear systems of equations is a very frequent operation and
thus important in many fields. The complexity using classical methods increases
linearly with the size of equations. The HHL algorithm proposed by Harrow et
al. achieves exponential acceleration compared with the best classical
algorithm. However, it has a relatively high demand for qubit resources and the
solution $\left| x \right\rangle $ is in a normalized form. Assuming that the
eigenvalues of the coefficient matrix of the linear systems of equations can be
represented perfectly by finite binary number strings, three hybrid iterative
phase estimation algorithms (HIPEA) are designed based on the iterative phase
estimation algorithm in this paper. The complexity is transferred to the
measurement operation in an iterative way, and thus the demand of qubit
resources is reduced in our hybrid algorithms. Moreover, the solution is stored
in a classical register instead of a quantum register, so the exact
unnormalized solution can be obtained. The required qubit resources in the
three HIPEA algorithms are different. HIPEA-1 only needs one single ancillary
qubit. The number of ancillary qubits in HIPEA-2 is equal to the number of
nondegenerate eigenvalues of the coefficient matrix of linear systems of
equations. HIPEA-3 is designed with a flexible number of ancillary qubits. The
HIPEA algorithms proposed in this paper broadens the application range of
quantum computation in solving linear systems of equations by avoiding the
problem that quantum programs may not be used to solve linear systems of
equations due to the lack of qubit resources.
