De Sitter Space as a Tensor Network: Cosmic No-Hair, Complementarity, and Complexity

TL;DR

This work investigates a de Sitter–MERA correspondence as a framework to study quantum gravity on super-Hubble scales, linking horizon thermodynamics, causal patches, and cosmological evolution to a tensor-network description. It identifies a quantum-channel fixed point that mirrors a cosmic no-hair behavior and analyzes complementarity through weak and strong perspectives, including a proposed SCMERA for local descriptions with horizon degrees of freedom. The paper also connects circuit complexity in the MERA to the de Sitter action, establishing a complexity–action bound that depends on the chosen gate set, and discusses symmetry-breaking limitations and potential generalizations to more realistic cosmologies. Overall, it provides an information-theoretic lens on de Sitter physics, offering a path toward integrating holography, error correction, and complexity into cosmic evolution models.

Abstract

We investigate the proposed connection between de Sitter spacetime and the MERA (Multiscale Entanglement Renormalization Ansatz) tensor network, and ask what can be learned via such a construction. We show that the quantum state obeys a cosmic no-hair theorem: the reduced density operator describing a causal patch of the MERA asymptotes to a fixed point of a quantum channel, just as spacetimes with a positive cosmological constant asymptote to de Sitter. The MERA is potentially compatible with a weak form of complementarity (local physics only describes single patches at a time, but the overall Hilbert space is infinite-dimensional) or, with certain specific modifications to the tensor structure, a strong form (the entire theory describes only a single patch plus its horizon, in a finite-dimensional Hilbert space). We also suggest that de Sitter evolution has an interpretation in terms of circuit complexity, as has been conjectured for anti-de Sitter space.

Figures (12)

• Figure 1: A periodic binary MERA. The green triangles denote the isometries and the blue squares denote the disentanglers. The kets labeled $|0\rangle$ are ancilla states inserted into each isometry. The action of the circuit is to take a state at the top and evolve it downward. In anticipation of the connection to de Sitter, the fine-graining direction is labelled as the direction of increasing $t$.
• Figure 2: The Penrose diagram of global (1+1)-dimensional de Sitter spacetime. As this is a spacetime diagram, time now runs from bottom to top. The boundaries of two complete disjoint causal patches, one centered at $\theta = 0$ and the other centered at $\theta = \pi$, are drawn with a dashed line, and the interiors of the patches are shaded. Light rays travel along $45^\circ$ lines in this diagram.
• Figure 3: A geometric de Sitter-MERA correspondence, mapping the MERA circuit to the top half of the de Sitter geometry. Note that the fine-graining direction of the MERA in this diagram points upward to match the future direction in the Penrose diagram. The domain of dependence of any pair of adjacent sites in the initial layer of the MERA is entirely contained within a single static patch in de Sitter. Two of the four possible static patch interiors are shaded in red. (The other two static patches are centered at $\theta = \pi/2$ and $\theta = 3\pi/2$.)
• Figure 4: A ternary MERA. Ancillae are suppressed in this diagram. A stationary causal cone with three sites per layer is indicated by the shaded region.
• Figure 5: (a) A single step of the MERA within the causal patch, viewed as a channel $\mathcal{E}$, and (b) the equivalent circuit diagram. Time runs in the downward direction in (a).

Theorems & Definitions (6)

• Definition 2.1
• Definition 2.2
• Theorem 3.1: Wald
• Example A.1
• Example A.2
• Proposition A.3