## Description

A level-\(\nu\) quantum divisible code is a CSS code whose \(X\)-type stabilizers form a \(\nu\)-even linear binary code in the symplectic representation and which admits a transversal gate at the \(\nu\)th level of the Clifford hierarchy. A CSS code is doubly-even (triply-even) if all \(X\)-type stabilizer generators have weight divisible by four (eight), i.e., if they form a doubly-even (triply-even) linear binary code. Doubly-even codes can yield a transversal \(S\) gate, while triply-even codes yield a transversal \(T\) gate for odd \(n\) [4].

More formally [5,6], a quantum divisible code is a CSS code defined from two linear binary codes \(C_{1,2}\) such that all weights in \(C_2\) share a common divisor \(\Delta > 1\), and all weights in each coset of \(C_2\) in \(C_1\) are congruent to \(\Delta\).

For example [6], if \(C_2\) is the first-order Reed-Muller code, and \(C_1/ C_2\) consists of quadratic forms with a bounded rank, then \([[n = 2m − 1, 1 \leq k \leq 1 + \sum_{i=1}^{m-4}(m − i), d = 3]]\) is a family of quantum divisible codes.

## Transversal Gates

## Gates

## Fault Tolerance

## Realizations

## Parent

- Generalized quantum divisible code — Generalized level-\(\nu\) quantum divisible codes reduce to quantum level-\(\nu\) divisible codes when \(t\) is a vector with \(\pm 1\) entries. The classical code formed by their \(X\)-type stabilizer generator matrix is \(\nu\)-even [9; pg. 10]. Both types of codes realize transversal gates outside of the Clifford group.

## Children

- \([[2^r-1, 1, 3]]\) quantum Reed-Muller code — \([[2^r-1, 1, 3]]\) quantum Reed-Muller codes come from RM\((1,m=r)\) codes, which are \((r-1)\)-even [10,11], and admit transversal gates at levels of the Clifford hierarchy. Building a tower of generalized divisible codes by starting with the Steane code yields the \([[2^r-1, 1, 3]]\) quantum Reed-Muller codes [9; Sec. VI.B].
- \([[3k + 8, k, 2]]\) triorthogonal code
- \([[49,1,5]]\) triorthogonal code

## Cousins

- Divisible code — The \(X\)-type stabilizers of a level-\(\nu\) quantum divisible code form a \(\nu\)-even linear binary code.
- Reed-Muller (RM) code — Quantum divisible codes can be constructed out of first-order RM codes [6].
- Concatenated quantum code — A fault-tolerant \(T\) gate on the five-qubit or Steane code can be obtained by concatenating with particular quantum divisible codes [6].
- Five-qubit perfect code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes [6].
- \([[7,1,3]]\) Steane code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes [6].
- Doubled color code — Doubled color codes are subsystem codes constructed using a generalization of the doubling transformation [12] that combines doubly-even linear binary codes to make triply-even codes. The doubling transformation is a special case of level lifting (from two to three) [9; Sec. VI.D].

## References

- [1]
- A. J. Landahl and C. Cesare, “Complex instruction set computing architecture for performing accurate quantum \(Z\) rotations with less magic”, (2013) arXiv:1302.3240
- [2]
- E. T. Campbell and M. Howard, “Unified framework for magic state distillation and multiqubit gate synthesis with reduced resource cost”, Physical Review A 95, (2017) arXiv:1606.01904 DOI
- [3]
- J. Haah and M. B. Hastings, “Codes and Protocols for DistillingT, controlled-S, and Toffoli Gates”, Quantum 2, 71 (2018) arXiv:1709.02832 DOI
- [4]
- S. Bravyi and A. Cross, “Doubled Color Codes”, (2015) arXiv:1509.03239
- [5]
- J. Hu, Q. Liang, and R. Calderbank, “Designing the Quantum Channels Induced by Diagonal Gates”, Quantum 6, 802 (2022) arXiv:2109.13481 DOI
- [6]
- J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176
- [7]
- C. Vuillot and N. P. Breuckmann, “Quantum Pin Codes”, IEEE Transactions on Information Theory 68, 5955 (2022) arXiv:1906.11394 DOI
- [8]
- P. A. Mishchenko and K. Xagawa, “Secure multi-party quantum computation based on triply-even quantum error-correcting codes”, (2022) arXiv:2206.04871
- [9]
- J. Haah, “Towers of generalized divisible quantum codes”, Physical Review A 97, (2018) arXiv:1709.08658 DOI
- [10]
- R. J. McEliece, “On periodic sequences from GF(q)”, Journal of Combinatorial Theory, Series A 10, 80 (1971) DOI
- [11]
- R. J. McEliece, “Weight congruences for p-ary cyclic codes”, Discrete Mathematics 3, 177 (1972) DOI
- [12]
- K. Betsumiya and A. Munemasa, “On triply even binary codes”, Journal of the London Mathematical Society 86, 1 (2012) arXiv:1012.4134 DOI

## Page edit log

- Jingzhen Hu (2022-05-04) — most recent
- Victor V. Albert (2022-05-04)

## Cite as:

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