# Quantum divisible code[1]

## Description

Consider a CSS code whose \(Z\)-stabilizers are determined by the dual of a classical \([n, k_1]\) linear binary code \(C_1\), and whose \(X\)-stabilizers are determined by a classical \([n, k_2]\) binary code \(C_2 \subset C_1\). This code is quantum divisible if 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, 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.

## Protection

## Gates

## Fault Tolerance

## Parent

## Cousins

- Divisible code — Quantum divisible codes are constructed via the CSS construction using a divisible linear binary code.
- Reed-Muller (RM) code — Quantum divisible codes can be constructed out of first-order RM codes.
- Quantum Reed-Muller code — Quantum RM codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.
- Triorthogonal code — Triorthogonal codes can be derived using a procedure that yields sufficient conditions for a CSS code to admit a given transversal diagonal logical gate. Quantum divisible codes are derived in a similar procedure, but one that yields necessary and sufficient conditions.
- 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.
- Five-qubit perfect code — A fault-tolerant \(T\) gate on the five-qubit code can be obtained by concatenating with particular quantum divisible codes.
- \([[7,1,3]]\) Steane code — A fault-tolerant \(T\) gate on the Steane code can be obtained by concatenating with particular quantum divisible codes.

## References

- [1]
- J. Hu, Q. Liang, and R. Calderbank, “Divisible Codes for Quantum Computation”, (2022) arXiv:2204.13176

## 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