MERA Tensor Network
A quantum state has too many parameters to write down. Take the lattice on the bottom row, add a layer that pairs neighbors via "isometries" (coarse-graining) and "disentanglers" (removing local correlations). Stack layers. The resulting tree-graph encodes a critical state efficiently — and its geometry is hyperbolic. Brian Swingle's 2009 observation: MERA looks exactly like the AdS/CFT dictionary. Maybe spacetime is built from entanglement.
physical qubits (boundary)
disentangler (square)
isometry (triangle)
causal wedge of region
why geometry emerges from quantum entanglement
A many-body quantum state on N sites lives in a Hilbert space of dimension d^N. For N = 100 spin-½ sites that's 2¹⁰⁰ amplitudes — more than atoms in the observable universe. Most states are useless: they have too much entanglement to ever arise from local physics.
Tensor networks (Vidal 2007, Cirac–Verstraete 2008) parameterize states by a graph of small tensors, each contracted with neighbors. The graph topology limits how entanglement can spread, restricting to physically reasonable states. Matrix Product States (MPS) handle 1D gapped phases. MERA (Multi-scale Entanglement Renormalization Ansatz, Vidal 2008) handles 1D critical states with log-violating entanglement.
The MERA graph: a hierarchical tree-with-extra-edges. Going up one layer halves the number of sites: a coarse-graining (isometry) merges pairs of sites into single effective sites. Disentanglers first remove short-range entanglement so the coarse-graining is unitary. Iterate to a single point — the "top."
Geometry: layer k has 2^(−k) of the original sites. The natural metric between sites is the graph distance through the network. Doing the geometry: this is the hyperbolic plane. The MERA tree literally lives in negatively curved space, with the boundary at the bottom row.
Ryu-Takayanagi: holographic duality says entanglement entropy of a boundary region A equals the area of the minimal surface in the bulk that ends on ∂A:
S(A) = Area(γ_A) / 4G_N
Compute this for MERA: count the network edges cut by a minimal "geodesic" surface dropping from A into the tree. The answer matches the actual entropy of A: S(A) ~ (c/3) log L for a 1D CFT of central charge c. The network reproduces the holographic entropy formula.
Swingle's insight (2009): the MERA is a discretization of AdS/CFT. The bulk (interior of the tree) is the anti-de Sitter geometry. The boundary is the conformal field theory. Quantum entanglement builds the radial direction. Cut the network — you cut spacetime in two.
Where this goes: Maldacena & Susskind ER=EPR conjecture (2013) — entangled qubits are connected by tiny wormholes. Van Raamsdonk: removing entanglement disconnects spacetime. The "It from Qubit" program reverse-engineers gravity from entanglement structure. We may live in a tensor network.