Abstract: We consider the problem of solving a special quartic-quadratic optimization problem with a single sphere constraint, namely, finding a global minimizer of 1/2 z*Az+β/2 sum|z_k|4 such that z*z=1. This problem spans multiple domains including quantum mechanics and chemistry sciences. We investigate the geometric properties of this optimization problem. When the matrix in the quadratic term is diagonal, the problem has unique local minima up to a phase shift and the problem can be solved in O(n log n)time. When the matrix in the quadratic term is rank-one, the global minima of the problem are unique under certain phase shift schemes. The strict-saddle property, which can imply polynomial time convergence of second-order algorithms, is proved when the coefficient of the quartic term is either at least O(n^1.5) or not larger than O(1). Finally, the Kurdyka-Lojasiewicz Exponent of this problem is estimated to be 1/4 and O(t^-0.5) local convergence rate is derived under some conditions. 

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