Holographic codes seen through ZX-calculus

TL;DR

This work reframes the pentagon holographic code in ZX-calculus terms, enabling a diagrammatic view of stabilisers, logical operators, and entanglement. It proves its encoding structure via six-qubit graph states, computes entanglement measures through ZX-simplifications, and connects to the Ryu-Takayanagi minimal-cut picture. Building on this, the authors introduce a dual {4,5} ZX-holographic code and investigate its decoding under erasure and depolarising Pauli noise using belief propagation and ordered statistics decoding, showing gauge fixing can induce sub-threshold performance. The results demonstrate that ZX-calculus provides a practical design and analysis toolkit for holographic quantum error-correcting codes and suggests avenues for constructing new codes with favorable decoding properties and spacetime interpretations.

Abstract

We re-visit the pentagon holographic quantum error correcting code from a ZX-calculus perspective. By expressing the underlying tensors as ZX-diagrams, we study the stabiliser structure of the code via Pauli webs. In addition, we obtain a diagrammatic understanding of its logical operators, encoding isometries, Rényi entropy and toy models of black holes/wormholes. Then, motivated by the pentagon holographic code's ZX-diagram, we introduce a family of codes constructed from ZX-diagrams on its dual hyperbolic tessellations and study their logical error rates using belief propagation decoders.

Figures (13)

• Figure 1: Encoding rate of the ZX-diagram equivalent to the pentagon holographic code.
• Figure 2: A pentagon holographic code ZX-diagram with $n=3$ layers.
• Figure 3: Logical recovery probability of the central bulk qubit under an erasure error model, decoded with the peeling decoder.
• Figure 4: Wormhole from Pastawski:2015qua and one its Pauli webs, spanning both sides of the wormhole.
• Figure 5: Logical error rate of the central bulk qubit for the $\{4,5\}$ ZX-holographic code without gauge fixing, decoded using belief propagation, under erasure errors at the boundary.