The linear quadratic (LQ) synchronization problem for multiagent systems is solved by developing a distributed algorithm. It is the first time in the literature to formulate the LQ synchronization problem consisting of auxiliary synchronization state variables in discrete time with a finite horizon. The solution to this LQ synchronization problem is first considered in a centralized setting by leveraging connections to an alternating direction method of multiplier and then extended to a distributed setting, in which the individual agents control inputs can be computed independently, thereby making the solution scalable. Numerical examples for both homogeneous and heterogeneous multiagent systems are given to demonstrate the effectiveness of the proposed method.