Pastawski, Yoshida, Harlow, and Preskill introduced the HaPPY code in their (now famous) paper, arXiv:1503.06237, as a way to model the AdS/CFT correspondence as a quantum error-correcting code. Although it is not explicitly mentioned in this paper, it seems (at least to me) that they are implying that HaPPY code is a certain type of MERA tensor network (remember that as Swingle showed, a MERA can model AdS). However, it is not clear to me how to explicitly relate the HaPPY code to a general MERA; more explicitly, I want to identify the disentanglers and isometries in the pentagon code introduced by HaPPY. In addition, the role of perfect tensors in HaPPY code is to make this code an exact quantum error-correcting code but, would a general MERA be able to model an approximate quantum error-correcting code?


The HaPPY code (https://arxiv.org/abs/1503.06237) is certainly inspired by MERA like tensor networks which were as you mentioned shown to be related to sliced AdS in a discrete sense. One can follow Swingle's paper (https://arxiv.org/abs/0905.1317) and https://arxiv.org/abs/1309.6282 for initial arguments. I am mentioning them here for completeness and readability.

A brief history behind the HaPPY code can be found on page 34 of https://arxiv.org/abs/1802.01040

The 1503.06237 paper was inspired by earlier work of Almheiri, Dong, and Harlow (https://arxiv.org/abs/1411.7041) where the authors argued that the CFT is a group of quantum error-correcting codes which correctly captures/explains the bulk reconstruction via AdS/CFT conjecture.

The motivation from MERA does not, however, mean that one can map the ingredients of MERA (isometries, disentanglers) with those in HaPPY code. But, it turns out the perfect tensors used in the HaPPY code are absolutely maximally entangled (AME) are always proportional to unitary or isometry. So there is one part of the resemblance. Generally, in this code physical variables on the boundary are encoded variables in the bulk. The encoder is a tensor network containing this information.

I am not sure about how general MERA can model an approximate quantum error-correcting code and other details right now. I will edit my answer in the future or wait to hear from some expert.


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