  In this paper, we first construct thermofield double states for bosonic
string theory in the light-cone gauge. We then obtain a coherent-thermal string
state and a thermal-coherent string state. We use the covariance matrix
approach to calculate the circuit complexity of coherent-thermal string states.
In this approach, we generate the optimal geodesics by a horizontal string
generator, and then obtain the circuit complexity using the length of the
minimal geodesics in the group manifold.
