Towards complexity in de Sitter space from the double-scaled Sachdev-Ye-Kitaev model

TL;DR

This work investigates how to define and interpret quantum complexity in de Sitter space from a microscopic holographic perspective based on a pair of doubled DSSYK models, their Liouville-dS2 CFT duals, and the SdS3 bulk. It develops and analyzes four complexity notions—spread, Krylov, query, and Nielsen—each with explicit microscopic definitions and bulk interpretations via entangler networks, operator growth, fusion/junction rules, and geodesic/one-shot circuit pictures. The results reveal that spread complexity counts entangled chord states and maps to antipodal time differences; Krylov complexity exhibits exponential growth consistent with maximal chaos across all dual descriptions; query complexity corresponds to the combinatorics of multipoint correlators and Wilson-line junctions; and a bi-invariant Nielsen count yields linear growth with partial JT-limit connections. Together these findings establish a concrete framework to translate information-theoretic measures into geometric and topological bulk quantities in dS holography, offering new avenues to quantify complexity and entanglement in cosmological spacetimes.

Abstract

How can we define complexity in dS space from microscopic principles? Based on recent developments pointing towards a correspondence between a pair of double-scaled Sachdev-Ye-Kitaev (DSSYK) models/ 2D Liouville-de Sitter (LdS$_2$) field theory/ 3D Schwarzschild de Sitter (SdS$_3$) space in arXiv:2310.16994, arXiv:2402.00635, arXiv:2402.02584, we study concrete complexity proposals in the microscopic models and their dual descriptions. First, we examine the spread complexity of the maximal entropy state of the doubled DSSYK model. We show that it counts the number of entangled chord states in its doubled Hilbert space. We interpret spread complexity in terms of a time difference between antipodal observers in SdS$_3$ space, and a boundary time difference of the dual LdS$_2$ CFTs. This provides a new connection between entanglement and geometry in dS space. Second, Krylov complexity, which describes operator growth, is computed for physical operators on all sides of the correspondence. Their late time evolution behaves as expected for chaotic systems. Later, we define the query complexity in the LdS$_2$ model as the number of steps in an algorithm computing n-point correlation functions of boundary operators of the corresponding antipodal points in SdS$_3$ space. We interpret query complexity as the number of matter operator chord insertions in a cylinder amplitude in the DSSYK, and the number of junctions of Wilson lines between antipodal static patch observers in SdS$_3$ space. Finally, we evaluate a specific proposal of Nielsen complexity for the DSSYK model and comment on its possible dual manifestations.

Figures (10)

• Figure 1: Left: Example of disk chord diagram, where we label the different levels (cyan) before each vertex (black dot), where $0$ (orange) represents the level where we will cut the diagram. Right: The chord diagram is sliced open (each level is represented with a dashed line). Each chord is a Wick contraction between the nodes (black dots) corresponding to the Hamiltonians in (\ref{['eq:chord tech']}), which can then end on the subsequent levels.
• Figure 2: Two ways to end with $l$ open chords after vertex $i$. Left: $l-1$ open chords before vertex $i$. Right: $l+1$ open chords before vertex $i$.
• Figure 3: Wilson line network (labeled $W_N$) on a global time slice in pure AdS$_3$ space. The Wilson lines (red lines) have been junctioned together (black dots) according to the rule (\ref{['eq:fusions']}) and amputated (blue dots) along a cutoff surface in the bulk interior, which is not necessarily at a constant radial location.
• Figure 4: Nielsen's geometric approach to operator complexity. The group manifold of unitary operators (white blob) is approximated as a smooth region. Left: A discrete set of elementary gates in a circuit (represented by orange dots) connecting the operators $\mathbb{1}$ and $x\in\;$SU(n) (cyan dots) is approximated through a continuous curve c($s$) (cyan). Right: Nielsen operator complexity picks the minimal length geodesic (blue) among all (cyan) of those connecting $\mathbb{1}$ and $x$.
• Figure 5: Illustration of spread complexity $\mathcal{C}_{\rm S}(t)$ (\ref{['eq:spread complexity']}) in $\mathcal{H}\otimes\mathcal{H}$. It maps the operator $\hat{\mathbb{N}}$ that counts the number of entangled chord states through $\mathcal{E}$ which are projected onto the state $\mathcal{O}_{E_0}\ket{0,~0}$ where $\mathcal{O}_{E_0}$ is defined in (\ref{['eq:O E0']}). See Sec. \ref{['sec:conclusions']} for comments on the DSSYK model and tensor networks.