nLab HaPPY code



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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:

From Harlow 18
  1. picking a perfect tensor TT of rank 6;

  2. picking a finite cutoff of the pentagonal tesselation {5,4}\{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 TT 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.


Tha HaPPY code is due to

following a precursor observation in

Review in:

See also:

  • Elliott Gesteau, Monica Jinwoo Kang, The infinite-dimensional HaPPY code: entanglement wedge reconstruction and dynamics (arXiv:2005.05971)

Last revised on February 28, 2023 at 08:28:09. See the history of this page for a list of all contributions to it.