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.
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