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## "Building manifolds from quantum codes", Michael Freedman, Matthew Hastings, 2021

*Reviewed March 15, 2024*

*Citation:* Freedman, Michael, and Matthew Hastings. "Building manifolds from quantum codes." Geometric and Functional Analysis 31.4 (2021): 855-894.

*Web:* https://link.springer.com/article/10.1007/s00039-021-00567-3

*Tags:* Mathematical, Higher-dimensional, Error-correcting-codes

This paper shows how to construct 11-dimensional Riemannian manifolds
from quantum error correcting codes with a "sparse lifting" condition.
The point of this paper is that it is a converse construction to a
more simple theme in quantum error correcting code theory. Given a
celluated manifold, one can choose any three-term sequence in its
cellular chain complex. This three-term chain complex can then be taken modulo 2.
A Z2 chain complex is exactly the data needed to define a quantum error correcting code.
Going back up, one can ask whether every quantum error correcting code comes from
a manifold. The answer is "yes", and that manifold's dimension is bounded by 11.
It is believed that 11 is not optimal - Zhenghan Wang believes it can be brought down to 6,
though certainly no lower.

One fantastic observation is that by choosing
very clever quantum error correcting codes,
one can obtain very interesting manifolds.
In particular, using Z2 error correcting
codes with constant encoding rate and linear
code distance one gets a manifold with
power-law Z2 systolic freedom.

The earlier paper

> Freedman, Michael H., David A. Meyer, and Feng Luo. "Z2-systolic freedom and quantum codes." Mathematics of quantum computation. Chapman and Hall/CRC, 2002. 303-338.

used manifolds with polylogarithm Z2 systolic freedom to construct (at that time)
optimal error correcting codes. This shows an amazing interplay between
the theory of systoles on manifolds and distance on error correcting codes.