Holographic complexity and charged scalar fields

We construct a time-dependent expression of the computational complexity of a quantum system, which consists of two conformal complex scalar field theories in d dimensions coupled to constant electric potentials and defined on the boundaries of a charged anti-de Sitter black hole in (d + 1) dimensions. Using a suitable choice of the reference state, Hamiltonian gates, and the metric on the manifold of unitaries, we find that the complexity grows linearly for a relatively large interval of time. We also remark that for scalar fields with very small charges the rate of variation of the complexity cannot exceed a maximum value known as the Lloyd bound.