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We introduce a method to solve the MaxCut problem efficiently based on quantum imaginary time evolution (QITE). We employ a linear Ansatz for unitary updates and an initial state that involve no entanglement. We apply the method to graphs with number of vertices |V | = 4, 6, 8, 10 and show that after ten QITE steps, the average solution is 100%, 99%, 98%, 97%, respectively, of the maximum MaxCut solution. By employing an imaginary-time-dependent Hamiltonian interpolating between a given graph and a subgraph with two edges excised, we show that the modified algorithm has a 100% performance converging to the maximum solution of the MaxCut problem for all graphs up to eight vertices as well as about 100 random samples of ten-vertex graphs. This improved method has an overhead which is polynomial in the number of vertices.

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