Abstract
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.
Highlights
The holographic principle—the idea that certain theories of gravity are dual to lower dimensional quantum field theory— has had wide-ranging applications within theoretical physics
We have studied the intersection of stabilizer states and fermionic Gaussian states, both efficiently describable classes of quantum states with a wide range of applications in quantum information theory and both condensedmatter and high-energy physics
We have introduced a graphical formalism for describing Majorana dimer states, free fermionic states characterized by entangled Majorana modes
Summary
The holographic principle—the idea that certain theories of gravity are dual to lower dimensional quantum field theory— has had wide-ranging applications within theoretical physics. The basis of this work is the tensor network construction of the hyperbolic pentagon code (HyPeC), a class of holographic models often named HaPPY codes after the authors’ initials [5] These codes explicitly realize holographic quantum error correction [3] by providing an error-correctable mapping from bulk to boundary degrees of freedom, reproducing many of the features of AdS/CFT. We show that the contraction of dimer-based tensor networks is equivalent to combining entangled Majorana pairs, replacing the computational difficulties of contraction by simple rules on dimer diagrams. This graphical language directly visualizes parities, physical correlations, and the entanglement structure of quantum states spanning the entire fermionic Hilbert space. Our work is an important step toward integrating discrete tensor network models of AdS/CFT into a unified setting
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