# Quantum error correction and large $N$

### Submission summary

 As Contributors: Alexey Milekhin Arxiv Link: https://arxiv.org/abs/2008.12869v2 (pdf) Date submitted: 2021-07-17 01:14 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:
Has been resubmitted

### Submission & Refereeing History

Resubmission 2008.12869v3 on 16 September 2021

Submission 2008.12869v2 on 17 July 2021

## Reports on this Submission

### Anonymous Report 1 on 2021-8-12 (Invited Report)

• Cite as: Anonymous, Report on arXiv:2008.12869v2, delivered 2021-08-12, doi: 10.21468/SciPost.Report.3384

### Report

I think this is a terrific and important paper. The work is extremely well-motivated: despite all the talk about error correcting codes and holography, before this work it was not at all clear how to see a priori that a large-$N$ gauge theory has something to do with error correction. Indeed one might have thought that the intricate and horrible details of the particular hamiltonians admitting simple gravity duals (e.g. a sparse spectrum) might have been required. But this paper makes it very clear that none of that is necessary, and that the requisite error correction property follows purely from kinematics of large $N$ singlets.

Furthermore, the paper gives a quantitative analysis of how the error correction property breaks down at finite $N$.
Since this property seems to be a deep fact about gravity, I believe this result will be quite important.
Indeed the main result of the paper implies that the error correction property transcends semiclassical gravity and applies to any semiclassical string theory.

The paper is also well written, with a very readable and compelling introduction.

One vague complaint, which actually can be blamed on ref [27]: there is essentially no such thing as "gauge symmetry", only gauge redundancy, and the presence of such a redundancy is not a physical property (for example, different dual descriptions of the same system need not have the same gauge group). 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.

Having said all these positive things I must admit that I have a confusion about the main premise.
The basic effect demonstrated here is that non-gauge invariant states (at least in these models with adjoint fermions) are approximately orthogonal at large $N$.
If, as in the toric code, we regard such "charged" (or "colored") excitations as errors, the fact that they are orthogonal means that different errors can be distinguished and therefore corrected.
The toric code is a UV completion of a (discrete) gauge theory, where such failures of gauge invariance occur only at high energy.
From this point of view of emergent gauge theory, the notion of the singlet sector as an error-correcting code makes perfect sense.
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.

-- "spacial" should be "spatial"
-- "so the are not dynamical"
should be "so they are not dynamical"
-- "can be draw from one fixed set of indices"
should be "can be drawn from one fixed set of indices"

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