The integration of phasor measurement units (PMUs) in power grids can greatly enhance the robustness of power grid state estimations. Due to the cost of components and installations, the number of PMUs is usually much less than that of buses in a power system. Therefore one of the critical problems faced by power system design is PMU placement, that is, identifying the buses on which the PMU should be installed. The objective of this paper is to develop PMU placement algorithms to improve the power grid state estimation. Unlike many existing PMU placement algorithms developed based on the concept of critical measurements, we use the estimation mean squared error (MSE) as the design metric. By applying a linear minimum MSE (MMSE) algorithm, the MSE is expressed as an explicit function of the locations of the PMUs. The problem is formulated as a combinatorial optimization problem that is known to be NP-hard. To balance the tradeoff between complexity and performance, we propose two low complexity algorithms, a greedy algorithm that sequentially searches for the best PMU location, and a heuristic ordered MSE algorithm that places PMUs at buses with highest MSE. Simulation results show that the proposed low complexity algorithms can almost achieve the globally optimum performance, and they significantly outperform existing PMU placement algorithms.