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Paper 1

Real-space RG, error correction and Petz map

Keiichiro Furuya, Nima Lashkari, Shoy Ouseph

Year

2020

Journal

arXiv preprint

DOI

arXiv:2012.14001

arXiv

2012.14001

There are two parts to this work: First, we study the error correction properties of the real-space renormalization group (RG). The long-distance operators are the (approximately) correctable operators encoded in the physical algebra of short-distance operators. This is closely related to modeling the holographic map as a quantum error correction code. As opposed to holography, the real-space RG of a many-body quantum system does not have the complementary recovery property. We discuss the role of large $N$ and a large gap in the spectrum of operators in the emergence of complementary recovery. Second, we study the operator algebra exact quantum error correction for any von Neumann algebra. We show that similar to the finite dimensional case, for any error map in between von Neumann algebras the Petz dual of the error map is a recovery map if the inclusion of the correctable subalgebra of operators has finite index.

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Paper 2

Tradeoffs on the volume of fault-tolerant circuits

Anirudh Krishna, Gilles Zémor

Year

2025

Journal

arXiv preprint

DOI

arXiv:2510.03057

arXiv

2510.03057

Dating back to the seminal work of von Neumann [von Neumann, Automata Studies, 1956], it is known that error correcting codes can overcome faulty circuit components to enable robust computation. Choosing an appropriate code is non-trivial as it must balance several requirements. Increasing the rate of the code reduces the relative number of redundant bits used in the fault-tolerant circuit, while increasing the distance of the code ensures robustness against faults. If the rate and distance were the only concerns, we could use asymptotically optimal codes as is done in communication settings. However, choosing a code for computation is challenging due to an additional requirement: The code needs to facilitate accessibility of encoded information to enable computation on encoded data. This seems to conflict with having large rate and distance. We prove that this is indeed the case, namely that a code family cannot simultaneously have constant rate, growing distance and short-depth gadgets to perform encoded CNOT gates. As a consequence, achieving good rate and distance may necessarily entail accepting very deep circuits, an undesirable trade-off in certain architectures and applications.

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