Majorana dimers and holographic quantum error-correcting codes

Holographic quantum error-correcting codes have been proposed as toy models that describe key aspects of the anti-de Sitter/conformal field theory (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 among bulk geometry, entanglement, and quantum error correction in AdS/CFT and lays the foundation for new models of holography.

Figure 1

Continuous (left) and discretized (right) reconstruction of an AdS bulk operator $\mathcal{O}$ along two (causal) wedges $\mathcal{W}\left[A\right]$ and $\mathcal{W}\left[B\right]$ [3], leading to two boundary operators ${\mathcal{O}}_A$ and ${\mathcal{O}}_B$ with support on boundary regions $A$ and $B$. The AdS time slice is projected onto the Poincaré disk, with the AdS boundary corresponding to the black outer circle. The discretization is a $\{5,4\}$ tiling.

Figure 2

Iterative contraction of the HyPeC for fixed bulk inputs of $|\overline{0}{\rangle }_5$ state vectors, with a cutoff after 3 iterations. Each step involves contracting a further layer of tiles, starting from the center at $n=0$. The asymptotic boundary of the Poincaré disk is drawn as a black circle.

Figure 3

Left: A Majorana dimer pair in an infinitely large contraction of HyPeC tiles. The endpoints of both dimers meet at the asymptotic boundary, and thus the dimer pair can be drawn as a double geodesic. Right: Full contraction for a $\overline{0}$ input on all tiles, with all dimers pairing up.

Figure 4

Entanglement entropy scaling with the size $\ell$ of the subsystem block for successive iterations $n$ of the contraction. Dashed line is a logarithmic fit of the $n\to \infty$ limit.

Figure 5

The greedy algorithm: The boundary region $A$ is pushed into the bulk to a new region $A^{\prime }$ by removing pentagon tiles with three (top) and four (bottom) open edges. Each pentagon can be in an arbitrary local superposition of $\overline{0}$ and $\overline{1}$, shown as gray-shaded dimers.

Figure 6

Reducing boundary regions in the HyPeC with the greedy algorithm, for two boundary regions $A$ and $B$. $A$ reduces to the same geodesic ${\gamma }_A={\gamma }_{A^{\text{C}}}$ as its complement $A^{\text{C}}$, while $B$ does not. On the left-hand side, the corresponding “greedy wedge” is shaded in the same color as the boundary regions. The residual dimers are shaded in red.

Figure 7

Each tile in the HyPeC can act as a mapping of $1\to 4$ edges (left) or $2\to 3$ edge (right). Arrows distinguish between “input” (IR) and “output” (UV) edges. Dimers extended from previous tiles are drawn as dashed curves, while new ones are drawn as solid curves. The dimer parities depend on the actual logical input on the tile.

Figure 8

Iterative contraction of the HyPeC, with Majorana dimers belonging to decoupled fermionic subsystems drawn in different colors. The number of such subsystems increases with iteration number $n$.

Figure 9

Top: Inserting two (left) and four (right) $\overline{1}$ tiles in the HyPeC. Beyond the local dimer parity flips in each tile, pairs of $\overline{1}$ tiles are connected by $Z$ strings (red lines), which flip dimer parities nonlocally. The endpoints of these strings (red dots) set the local orientation of the $\overline{1}$ tiles connected to it. Bottom: Equivalent $Z$ string configuration of the upper two states.

Figure 10

Effective bit thread picture in the asymptotically large HyPeC: All dimers are paired along bulk geodesics and any boundary region $A$ has a maximal flow of bit threads (dimer pairs) through the Ryu-Takayanagi surface ${\gamma }_A$. The threads that pass ${\gamma }_A$, each contributing $log2$ to the entanglement entropy $S_A$, are highlighted. Note that both the number of such threads and $S_A$ are UV divergent.