Krylov spread complexity as holographic complexity beyond JT gravity

TL;DR

Understanding holographic complexity in black hole interiors remains a central challenge. This work extends the known complexity=volume duality from classical JT gravity to a finite-temperature, fully quantum regime using a duality between double-scaled SYK and sine dilaton gravity. It demonstrates that Krylov spread complexity in DSSYK precisely matches the bulk geodesic length in sine dilaton gravity at finite temperature, valid for all q in (0,1) and all β, by relating length to the boundary Krylov dynamics via the transfer matrix. The authors further identify the first quantum correction to the bulk length as a potential dual to bulk quantum-field Krylov complexity and discuss the switchback effect and future extensions to higher dimensions and non-perturbative completions.

Abstract

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 triple-scaled SYK at infinite temperature and the complexity = volume proposal in classical JT gravity. Our work utilizes the recent connection between double-scaled SYK 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. From the latter 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. Finally, we comment on the switchback effect, whose presence would make the Krylov spread complexity a fully fledged holographic complexity at least in sine-dilaton gravity.

Figures (2)

• Figure 1: Numerical values for the finite temperature Krylov spread complexity $C_K(t)_\beta$ at various temperatures $\beta$ and $q^2=0.6$ (solid). It is visible that for larger $\beta$, the initial onset at $t=0$ that accounts for higher complexity of the Euclidean state preparation is larger and the complexity growth slows down. The complexities are contrasted with the expectation value of the classical length in sine dilaton gravity at temperature $\beta$ (dashed), normalized by $2\left|\log q \right|$.
• Figure 2: Numerically evaluated Krylov complexity $C_K(t)_\beta$ (solid), normalized by the classical length $\langle \hat{L}\rangle_{\text{cl.}}$ (and a factor of $2\left|\log q \right|$ that is omitted in the label) for $\beta=1$ and various $q$ (color coding) vs. the leading order quantum correction $\langle \hat{L}\rangle_{q\text{-corr.}}$ (dot-dashed), also normalized by $\langle \hat{L}\rangle_{\text{cl.}}$, for the same values of $q$ and $\beta$. For $q$ close to one, the curves fall on top of each other, signaling perfect agreement; For $q$ further away from $q=1$, the deviation systematically increases because of higher order corrections that were not taken into account.