# Haah cubic code[1]

## Description

Class of stabilizer codes on a length-\(L\) cubic lattice with one or two qubits per site. We also require that the stabilizer group \(\mathsf{S}\) is translation invariant and generated by two types of operators with support on a cube.

In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival ''string operators'', meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase.

Haah showed in his original construction that there is exactly one non-CSS code of this form, and 17 CSS codes [1]. The non-CSS code is labeled code 0, and the rest are numbered from 1 - 17. Codes 1-4, 7, 8, and 10 do not have string logical operators [1,2]. Encodings using geometries with boundaries as well as lattice defects have been studied [3].

Straightforward generalizations of the above codes exist to modular qudits, oscillators, and rotors [4,5].

## Protection

## Threshold

## Parents

- Qubit stabilizer code
- Fracton code — Haah cubic codes are the first examples of Type-II fracton phases [6].

## Cousins

- Color code — The color and cubic code families both include 3D codes that do not admit string-like operators.
- Generalized surface code — The energy of any partial implementation of code 1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
- Lifted-product (LP) code — A lifted-product code for the ring \(R=\mathbb{F}_2[x,y,z]/(x^L-1,y^L-1,z^L-1)\) is the cubic code [7; Appx. B].
- Fibonacci code — The Fibonacci code is designed to mimic the fractal properties of (quantum) Haah cubic code so that studying the former can help us toward the development of an efficient algorithm for the latter.
- Self-correcting quantum code — Cubic code 1 is partially self-correcting with a logarithmic energy barrier [8].

## References

- [1]
- J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
- [2]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [3]
- C. T. Aitchison et al., “No Strings Attached: Boundaries and Defects in the Cubic Code”, (2023) arXiv:2308.00138
- [4]
- J. Haah, Two generalizations of the cubic code model, KITP Conference: Frontiers of Quantum Information Physics, UCSB, Santa Barbara, CA.
- [5]
- V. V. Albert, S. Pascazio, and M. H. Devoret, “General phase spaces: from discrete variables to rotor and continuum limits”, Journal of Physics A: Mathematical and Theoretical 50, 504002 (2017) arXiv:1709.04460 DOI
- [6]
- M. Pretko, X. Chen, and Y. You, “Fracton phases of matter”, International Journal of Modern Physics A 35, 2030003 (2020) arXiv:2001.01722 DOI
- [7]
- P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
- [8]
- S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI

## Page edit log

- Victor V. Albert (2022-01-11) — most recent
- Siddharth Taneja (2021-12-19)

## Cite as:

“Haah cubic code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/haah_cubic