Operator complexity for quantum scalar fields and cosmological perturbations

Abstract

We calculate the operator complexity for the displacement, squeeze and rotation operators of a quantum harmonic oscillator, assuming equal computational cost for the corresponding fundamental gates. The complexity of the time-dependent displacement operator is constant, equal to the magnitude of the coherent state parameter, while the complexity of unitary evolution by a generic quadratic Hamiltonian is proportional to the amount of squeezing and is sensitive to the time-dependent phase of the unitary operator. We apply these results to study the complexity of a free massive scalar field, finding that the complexity has a period of rapid linear growth followed by a saturation determined by the UV cutoff and the number of spatial dimensions. We also study the complexity of the unitary evolution of quantum cosmological perturbations in de Sitter space, which can be written as time-dependent squeezing and rotation operators on individual Fourier mode pairs. The complexity of a single mode pair at late times grows linearly with the number of $e$-folds, while the complexity at early times oscillates rapidly due to the sensitivity of operator complexity to the phase of unitary time evolution. Integrating over all modes, the total complexity of cosmological perturbations scales as the square root of the (exponentially) growing volume of de Sitter space, suggesting that inflation leads to an explosive growth in complexity of the Universe.