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Krylov spread complexity as holographic complexity beyond JT gravity
Michal P. Heller, Jacopo Papalini, Tim Schuhmann
TL;DR
The microscopic boundary interpretation of holographic complexity remains open, with only one quantitative match previously identified: infinite-temperature Krylov spread complexity in triple-scaled SYK and classical JT complexity = volume. Using the double-scaled SYK–sine-dilaton gravity duality, this paper extends the match to finite temperatures and the quantum regime at disk level.
Problem
The microscopic boundary interpretation of holographic complexity remains open, with only one quantitative match previously identified: infinite-temperature Krylov spread complexity in triple-scaled SYK and classical JT complexity = volume.
Source: p. 1Method
The authors use the double-scaled SYK–sine-dilaton gravity duality and its holographic dictionary to recast bulk length as Krylov spread complexity.
Source: pp. 1, 3Results
The Krylov spread complexity–sine-dilaton gravity length match holds at the quantum level for all allowed DSSYK q values and all β.
Source: pp. 3, 5Takeaways & Limitations
At finite temperature, matching Krylov complexity requires assigning complexity to Euclidean Hartle–Hawking state preparation as well as Lorentzian evolution.
Source: pp. 3, 5A proof that Krylov spread complexity exhibits the switchback effect remains outstanding, and the authors say it would complete its understanding as genuine holographic complexity.
Source: p. 5
Abstract
from arXiv · showhide
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.
SUPPLEMENTAL MATERIAL
The supplemental material develops the sine-dilaton gravity–DSSYK quantization dictionary, makes the complexity–volume relation an operator statement, and clarifies the SYK reference state and numerical procedure.
SM1: More details on sine dilaton gravity and the duality
- Canonical quantization yields the DSSYK transfer matrix, while global-slice Wheeler–DeWitt wavefunctions solve the associated difference equation and their orthogonality gives the DSSYK spectral density.After discretization, the wavefunctions reduce to q-Hermite polynomials.Source: SUPPLEMENTAL MATERIAL · p. 6
SM2: Analytic expression for the two-point function of a massive probe in sine dilaton gravity
- The calculation inserts two completeness relations and uses energy eigenvalues given by (22), with spectral density ρ(θ) = (q2, e±2ıθ; q2)∞ associated with wavefunctions (25), noting q-Hermite polynomials.The shorthand expands as (q2; q2)∞(e+2ıθ; q2)∞(e−2ıθ; q2)∞.Source: SUPPLEMENTAL MATERIAL · pp. 6–7
SM3: Complexity equals volume: operator statement
- In finite-temperature DSSYK, the Krylov operator equals the sine-dilaton wormhole length operator, identifying Krylov spread complexity with complexity equals volume.This identification applies for times shorter than exponential in the entropy and reproduces the explicit result in the main text.Source: SUPPLEMENTAL MATERIAL · p. 7
SM4: SYK interpretation of the chord Krylov problem
- The finite-temperature SYK Krylov problem uses an equal superposition of energy eigenstates as reference state and is equivalent to the chord problem after coupling averaging.The equivalence applies when complexity includes both Lorentzian evolution and Euclidean state preparation, rather than placing preparation in the reference state.Source: SUPPLEMENTAL MATERIAL · p. 7
SM5: Krylov spread complexity: Numerical evaluation
- The finite-temperature numerical evaluation uses kcut = 50 and selects ncut by requiring the truncated state’s norm to agree with one to the desired accuracy.The sums are truncated at sufficiently large cutoffs, with the k-sum reported to converge sufficiently fast.Source: SUPPLEMENTAL MATERIAL · pp. 7–8