The Holography of Spread Complexity: A Story of Observers

We propose a holographic description of spread complexity and its rate in 2D CFTs, building on the pioneering work \cite{Caputa:2024sux}. By exploiting the $SL(2,\mathbb{R})$ symmetry, we construct the Krylov basis and demonstrate that its non-uniqueness gives rise to ambiguities analogous to those in quantum complexity. Within the AdS/CFT correspondence, we identify the spread complexity as the energy measured by a bulk observer, with its rate corresponding to the radial momentum. These results suggest a novel holographic interpretation of complexity ambiguity: rather than arising from infinitely many gravitational observables, it may reflect the existence of infinitely many observers, each measuring complexity from their own perspective.