Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence

TL;DR

The paper develops a family of exactly solvable holographic quantum error-correcting codes built from perfect tensors on hyperbolic tilings to model the bulk/boundary correspondence in AdS/CFT.It provides an explicit encoding isometry from bulk to boundary, demonstrates exact RT-like entanglement behavior for connected regions, and shows how bulk operators can be reconstructed on multiple boundary regions, embodying AdS-Rindler reconstruction as QECC.The work further analyzes multipartite entanglement via greedy geodesics, introduces causal and entanglement wedges within the tensor-network setting, and explores erasure thresholds and stabilizer implementations, including black hole and wormhole toy models.Overall, it offers a concrete, solvable framework linking quantum error correction, tensor networks, and holography, while outlining key limitations and numerous open questions for extending the analogy.

Abstract

We propose a family of exactly solvable toy models for the AdS/CFT correspondence based on a novel construction of quantum error-correcting codes with a tensor network structure. Our building block is a special type of tensor with maximal entanglement along any bipartition, which gives rise to an isometry from the bulk Hilbert space to the boundary Hilbert space. The entire tensor network is an encoder for a quantum error-correcting code, where the bulk and boundary degrees of freedom may be identified as logical and physical degrees of freedom respectively. These models capture key features of entanglement in the AdS/CFT correspondence; in particular, the Ryu-Takayanagi formula and the negativity of tripartite information are obeyed exactly in many cases. That bulk logical operators can be represented on multiple boundary regions mimics the Rindler-wedge reconstruction of boundary operators from bulk operators, realizing explicitly the quantum error-correcting features of AdS/CFT recently proposed by Almheiri et. al in arXiv:1411.7041.

Figures (27)

• Figure 1: Diagrammatic tensor notation, here showing that $T$ is an isometry.
• Figure 2: Operator pushing through an isometric tensor.
• Figure 3: If $\mathcal{H}_A=\mathcal{H}_{A_2}\otimes \mathcal{H}_{A_1}$, then we can move one of the factors to the output while preserving the isometric structure.
• Figure 4: White dots represent physical legs on the boundary. Red dots represent logical input legs associated to each perfect tensor.
• Figure 5: A cut through a holographic tensor network by a curve $c$ bounded by $\partial A$. Boundary indices $a$ and $b$ are uncontracted in $A$ and its complement $A^c$ respectively; tensors $P$ and $Q$ are contracted by summing over the index $i$ which is cut by $c$.

Theorems & Definitions (18)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Definition 4
• Theorem 2
• Theorem 3
• Theorem 4
• proof
• Definition 5