A bulk manifestation of Krylov complexity

TL;DR

This work establishes a concrete AdS/CFT entry for Krylov (K) complexity by showing that boundary Krylov complexity of the infinite-temperature thermofield double in double-scaled SYK corresponds to the length of the two-sided wormhole in JT gravity. By mapping the Krylov basis to fixed-chord-number states via chord-diagram techniques, the authors relate boundary state growth to bulk geometric length in the triple-scaling, Liouville/JT regime. The result provides a precise, quantitative boundary–bulk dictionary for a nontrivial notion of quantum complexity, with KC growth matching the log-cosh evolution of bulk wormhole length and offering a controlled setting to probe holographic complexity beyond circuit-defined measures. These findings illuminate how low-dimensional holography encodes complexity in geometric data and suggest avenues for extending the KC–bulk correspondence to more general states and higher-dimensional analogs.

Abstract

There are various definitions of the concept of complexity in Quantum Field Theory as well as for finite quantum systems. For several of them there are conjectured holographic bulk duals. In this work we establish an entry in the AdS/CFT dictionary for one such class of complexity, namely Krylov or K-complexity. For this purpose we work in the double-scaled SYK model which is dual in a certain limit to JT gravity, a theory of gravity in AdS$_2$. In particular, states on the boundary have a clear geometrical definition in the bulk. We use this result to show that Krylov complexity of the infinite-temperature thermofield double state on the boundary of AdS$_2$ has a precise bulk description in JT gravity, namely the length of the two-sided wormhole. We do this by showing that the Krylov basis elements, which are eigenstates of the Krylov complexity operator, are mapped to length eigenstates in the bulk theory by subjecting K-complexity to the bulk-boundary map identifying the bulk/boundary Hilbert spaces. Our result makes extensive use of chord diagram techniques and identifies the Krylov basis of the boundary quantum system with fixed chord number states building the bulk gravitational Hilbert space.

Figures (12)

• Figure 1: Untangling a chord diagram.
• Figure 2: "Cutting open" a chord diagram Berkooz:2018jqr. The cut-open chord diagram can be thought of as a process in which one starts with zero open chords and ends with zero open chords, while chords are created and annihilated in between.
• Figure 3: Left: The surface $-T_1^2-T_2^2+X^2=-l_{AdS}^2$ embedded in 3-dimensional space. The wormhole length is represented by the red line, and the boundaries are represented by blue lines. The yellow dots represent the anchoring points at the boundary, between which the wormhole length is computed. The yellow lines represent the Schwarzschild horizon and the areas colored yellow represent the regions covered by the Schwarzschild coordinates. Right: Schematic Penrose diagram of the same geometry.
• Figure 4: The 15 chord diagrams contributing to $M_6$: the top row shows chord diagrams with zero intersections (5 diagrams), the second row shows chord diagrams with 1 intersection (6 diagrams), the third row shows diagrams with 2 intersections (3 diagrams) and the fourth row shows the only chord diagram with 3 intersections. These numbers correspond to the calculation of $M_6$ in (\ref{['First_three_even_moments']}).
• Figure 5: Lanczos coefficients in DSSYK for different values of $q$, denoted $b_n\equiv b(n,q)$. Left: Linear scale along both axes, focusing at small values of $n$. Right: Log-log plot. In this scale, the initial linear shape is compatible with a square-root behavior of $b_n$; figure \ref{['fig:Lanczos_example']} illustrates this in more detail.