Operator approach to complexity: Excited states

We evaluate the complexity of the free scalar field by the operator approach in which the transformation matrix between the second quantization operators of the reference state and target state is regarded as the quantum gate. We first examine the system in which the reference state is two noninteracting oscillators with the same frequency ${\omega }_0$ while the target state is two interacting oscillators with frequency ${\tilde{\omega }}_1$ and ${\tilde{\omega }}_2$. We calculate the geodesic length on the associated group manifold of gate matrix and reproduce the known value of ground-state complexity. Next, we study the complexity in the excited states. Although the gate matrix is very large we can transform it to a diagonal matrix and obtain the associated complexity. We explicitly calculate the complexity in several excited states and prove that the square of the geodesic length in the general state $|\mathrm{n},\mathrm{m}⟩$ is $D_{(\mathrm{n},\mathrm{m})}^2=(n+1){\left(\ln \sqrt{\frac{{\tilde{\omega }}_1}{{\omega }_0}}\right)}^2+(m+1){\left(\ln \sqrt{\frac{{\tilde{\omega }}_2}{{\omega }_0}}\right)}^2$. The results are extended to the $N$ couple harmonic oscillators which correspond to the lattice version of the free scalar field.