# Operator Approach to Complexity : Excited States

## TL;DR

This paper evaluates the complexity of free scalar fields using an operator approach, calculating the geodesic length for ground and excited states, and extends the results to lattice models of harmonic oscillators.

## Contribution

It introduces a method to compute complexity for excited states of scalar fields using operator transformations and geodesic lengths, extending to lattice models.

## Key findings

- Explicit formulas for excited state complexity are derived.
- Complexity in excited states depends on quantum numbers and frequency ratios.
- Results are generalized to N coupled harmonic oscillators.

## Abstract

We evaluate the complexity of the free scalar field by the operator approach in which the transformation matrix between the second quantization operators of reference state and target state is regarded as the quantum gate. We first examine the system in which the reference state is two non-interacting oscillators with 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 geodesic length in the general state $|{\rm n,m}\rangle$ is $D_{\rm (n,m)}^2={\rm (n+1)}\left(\ln {\sqrt{\tilde \omega_1\over \omega_0}}\,\right)^2 +{\rm (m+1)}\left(\ln {\sqrt{\tilde \omega_2\over \omega_0}}\,\right)^2$. The results are extended to the N couple harmonic oscillators which correspond to the lattice version of free scalar field.