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## "The fragility of quantum information?", Barbara Terhal, 2012

*Reviewed February 24, 2024*

*Citation:* Terhal, Barbara M. "The fragility of quantum information?." International Conference on Theory and Practice of Natural Computing. Berlin, Heidelberg: Springer Berlin Heidelberg, 2012.

*Web:* https://arxiv.org/abs/1305.4004

*Tags:* Philosophical, Expository, Error-correcting-codes, Higher-dimensional, Toric-code

In this essay Barbara Terhal gives her opinions about the question
of whether quantum information is inherently more fragile than classical
information. Her conclusion comes, unsurprisingly, from the theory
of quantum error correction. She believes that stable quantum information
in its most naive sense should correspond to self correcting
quantum codes. The toric code, and similar codes, are not self
correcting in 2D because the boundary of defect-creating
operators are 0-dimensional and hence do not scale with separation.
Even in 3D, at least one of the X or Z defect-creating operators will have 0-dimensional
boundary. Going to 4D, both the X and Z defect-creating operators will have 1-dimensional
boundary. This boundary scales with separation and hence there is an energy cost to
error propagation, and hence the code will be self-correcting.
Thus, the conclusion is that quantum information is inherently more fragile
but this wouldn't be the case if we lived in a 4D world.

Of course, this conclusion is stated with a myriad of caveats.
For instance, if you are able to make classical post selection then
you can enforce a better-than-nature cooling which confines defects.
This is how quantum error correction works on quantum computers,
which Terhal is confident we can make.

Another caveat is that maybe there are exotic 3D codes which don't
behave like the toric code, and which are self-correcting. The most promising
candidate is the use of fracal codes, which exploit the missing room between dimensions.
In particular, she is excited about Bravyi and Haah's code:

> Bravyi, Sergey, and Jeongwan Haah. "Energy landscape of 3D spin Hamiltonians with topological order." Physical review letters 107.15 (2011): 150504.

> Bravyi, Sergey, and Jeongwan Haah. "Analytic and numerical demonstration of quantum self-correction in the 3D cubic code." arXiv preprint arXiv:1112.3252 (2011).