Contents

# Contents

## Idea

The hyperbolic pentagon code or HaPPY code is a quantum error correction code (a class of such codes really, indexed by a “cutoff” natural number) which is thought to exhibit characteristic properties akin to the encoding of bulk-quantum states by boundary-states expected in the AdS/CFT correspondence. In particular, the HaPPY code (or rather the tensor network that defines it) exhibits a discretized form of the Ryu-Takayanagi formula for holographic entanglement entropy.

Concretely, the the HaPPY code subspace is the image of the linear map formed by:

1. picking a perfect tensor $T$ of rank 6;

2. picking a finite cutoff of the pentagonal tesselation $\{5,4\}$ of the hyperbolic plane;

3. regarding its Poincaré dual graph as a tensor network (string diagram in finite-dimensional vector spaces) by

1. assigning $T$ to each vertex at the center of the pentagons (show in blue), with 5 of its indices contracted with its neighbours in the hyperbolic plane,

2. and its 6th uncontracted index remaining as an input (shown in red);

3. regading the uncontrated edges at the cutoff boundary as output (shown in white)

and thus as a linear map form the tensor product over the bulk-vertices to the tensor product over the edges sticking out over the boundary.

## References

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following a precursor observation in

Review in: