  Holographic quantum error-correcting codes have been proposed as toy models
that describe key aspects of the AdS/CFT correspondence. In this work, we
introduce a versatile framework of Majorana dimers capturing the intersection
of stabilizer and Gaussian Majorana states. This picture allows for an
efficient contraction with a simple diagrammatic interpretation and is amenable
to analytical study of holographic quantum error-correcting codes. Equipped
with this framework, we revisit the recently proposed hyperbolic pentagon code
(HyPeC). Relating its logical code basis to Majorana dimers, we efficiently
compute boundary state properties even for the non-Gaussian case of generic
logical input. The dimers characterizing these boundary states coincide with
discrete bulk geodesics, leading to a geometric picture from which properties
of entanglement, quantum error correction, and bulk/boundary operator mapping
immediately follow. We also elaborate upon the emergence of the Ryu-Takayanagi
formula from our model, which realizes many of the properties of the recent bit
thread proposal. Our work thus elucidates the connection between bulk geometry,
entanglement, and quantum error correction in AdS/CFT, and lays the foundation
for new models of holography.
