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.

• Received 15 February 2019

DOI:https://doi.org/10.1103/PhysRevD.99.106013

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Published by the American Physical Society