Krylov Spread Complexity as Holographic Complexity beyond Jackiw-Teitelboim Gravity

One of the important open problems in quantum black hole physics is a dual interpretation of holographic complexity proposals. To date the only quantitative match is the equality between the Krylov spread complexity in the triple-scaled Sachdev-Ye-Kitaev (SYK) model and the complexity = volume proposal in classical Jackiw-Teitelboim gravity. Our Letter utilizes the recent connection between the double-scaled SYK model and sine dilaton gravity to show that the quantitative relation between Krylov spread complexity and complexity = volume extends to finite temperatures and to full quantum regime on the gravity side at disk level. Furthermore, we isolate the first quantum correction to the complexity = volume proposal and propose to view it as a complexity of quantum fields in the bulk. The key lesson from our Letter is that gravity demands to assign complexity also to Euclidean state preparation.