# Quantum error correction and large $N$

### Submission summary

 As Contributors: Alexey Milekhin Arxiv Link: https://arxiv.org/abs/2008.12869v3 (pdf) Date accepted: 2021-09-22 Date submitted: 2021-09-16 22:43 Submitted by: Milekhin, Alexey Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Quantum Physics Approach: Theoretical

### Abstract

In recent years quantum error correction(QEC) has become an important part of AdS/CFT. Unfortunately, there are no field-theoretic arguments about why QEC holds in known holographic systems. The purpose of this paper is to fill this gap by studying the error correcting properties of the fermionic sector of various large $N$ theories. Specifically we examine $SU(N)$ matrix quantum mechanics and 3-rank tensor $O(N)^3$ theories. Both of these theories contain large gauge groups. We argue that gauge singlet states indeed form a quantum error correcting code. Our considerations are based purely on large $N$ analysis and do not appeal to a particular form of Hamiltonian or holography.

###### Current status:
Publication decision taken: accept

Editorial decision: For Journal SciPost Physics: Publish
(status: Editorial decision fixed and (if required) accepted by authors)

Dear Referee,

>The statement that "QEC is tied to the presence of gauge symmetries" is therefore a problematic one. It would be >nice to know what exactly is its invariant meaning.
As far as I understand the results of ref. [27], it was the presence of gauge redundancy that allowed them to build precursor operators. In other, more invariant words, they linked the redundancy of quantum error correction to the gauge redundancy. I changed the corresponding statement in the Introduction as "It was suggested earlier
[27] that the redundancy in holographic QEC maybe tied the gauge redundancy."

>In the context of gauge theories with gravity duals, however, we usually regard the gauge invariance as an exact >redundancy, so that the physical hilbert space contains only gauge invariant states. In that context, I do not >understand a physical role for these states with errors.
This is a very good comment and I completely agree with it. From bigger perspective I see the current results just as a first step in understanding more general quantum error correcting properties. I have been thinking about more general setup, where errors are allowed to be singlets, but I have not made much progress.
As a "consolation prize", the results in the paper can be helpful for simple matrix models such as BFSS. Ref. [28] argued that ungauging SU(N) in BFSS preserves the bulk dual. Non-singlet states correspond to folded strings in the bulk. Also ref. [28] argued that such non-singlet excitations always have a large energy and so they are localized near the boundary. The argumentation was based on bulk locality and is not directly applicable to other matrix models without smooth bulk dual. However, ref. [29] observed numerically that in other "ungauged" matrix models non-singlets have high energy too. Results in current paper might suggest that this is a general phenomena at large N. However, I have not managed to formulate this more rigorously, as properties of near-orthogonality and being highly energetic are not necessarily related.

Best regards,
Alexey