*Citation:* Li, Zhi, and Latham Boyle. "The Penrose Tiling is a Quantum Error-Correcting Code." arXiv preprint arXiv:2311.13040 (2023).

*Web:* https://arxiv.org/abs/2311.13040

*Tags:* Error-correcting-codes

In this paper, the authors observe that the local indistinguishably property of aperiodic Penrose tilings can be used to create error correcting codes. The idea is brain-dead simple: you take your space of states to be all possible tilings, modulo Euclidean transformations. If an arbitrary finite connected region of your state is erased, then it can be recovered. Hence, this is an "error correcting code".

Note the plethora of obvious issues, though. This code can only recover errors made by erasure of a finite contiguous region. That's a pretty big restriction. Additionally, no protocols are given for how to form the Hilbert space from a lattice model or non-pathological continuous model. On top of this, a simple argument given in the conclusion shows that even if such a Hilbert space were created there would be no local Hamiltonian which would implement this code. Additionally, this code has no finite analogues. It must be played on an unphysically-large infinite grid.

All in all, it feels a bit deceitful to call this an error correcting code. On top of that, it feels even more deceitful to call it a quantum error correcting code. There's really nothing quantum about it. Every classical error correcting code can obviously be made quantum just be looking at the Hilbert space spanned by your classical information. The subtelty in making a quantum code is that you need to add Z-type checks to make sure phases can't be continuously deformed, but this isn't done in this paper and moreover it can't be done since the PT doesn't admit a local Hamiltonian. In other words, the authors made this code "quantum" for seemingly no reason.

Perhaps the reason that they made the connection was so that they could connect their non-error correcting code to quantum gravity. The entirety of the connection is that relativity is reference-frame independent, so it should correspond to a code where a superposition is made over possible reference frames, just as in this code. Seems very weak. We know that quantum gravity is sort of an error correcting code, which has some ties to aperiodic tilings, but this connection is misleading at best and disingenuous at worst. The most relevant reference is this one:

> Boyle, Latham, Madeline Dickens, and Felix Flicker. "Conformal quasicrystals and holography." Physical Review X 10.1 (2020): 011009.

This paper reminds me of Beni Yoshida's fractal codes,

> Yoshida, Beni. "Information storage capacity of discrete spin systems." Annals of Physics 338 (2013): 134-166.

The difference here is that Yoshida's codes are actually error correcting codes. Additionally, Yoshida recognizes that his code isn't fundamentally quantum and works with classical spin systems. In later works, one quantum-information theoretic properties of the corresponding phase seem to become relevant, the model is generalized and analyzed in the quantum case.