# Qubit code

## Description

## Protection

A convenient and often considered error set is the Pauli error or Pauli string basis. For a single qubit, this set consists of products of powers of the Pauli matrices \begin{align} X=\begin{pmatrix}0 & 1\\ 1 & 0 \end{pmatrix}\,\,\text{ and }\,\,Z=\begin{pmatrix}1 & 0\\ 0 & -1 \end{pmatrix}~. \end{align} For multiple qubits, error set elements are tensor products of elements of the single-qubit error set.

The Pauli error set is a unitary and Hermitian basis for linear operators on the multi-qubit Hilbert space that is orthonormal under the Hilbert-Schmidt inner product; it is a prototypical nice error basis [1]. The distance associated with this set is often the minimum weight of a Pauli string that implements a nontrivial logical operation in the code. The minimum weight of a Pauli error that has a non-zero expectation value for some code basis state is called the diagonal distance [2]. Codes whose distance is greater than the diagonal distance are degenerate.

## Decoding

## Notes

## Parent

## Children

## Cousins

- Fermionic code — While the Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via the Jordan-Wigner transformation [4], codes based on the two algebras have different notions of locality and thus qualitatively different physical interpretations.
- Fock-state bosonic code — Fock-state code whose codewords are finite superpositions of Fock states with maximum occupation \(N\) can be mapped into a qubit code with \(n\geq\log_2 N\) by performing a binary expansion of the Fock-state labels \(n\) and treating each binary digit as an index for a qubit state. Pauli operators for the constituent qubits can be expressed in terms of bosonic raising and lowering operators [5]. However, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [6].
- Group-based quantum code — Group quantum codes whose physical spaces are constructed using the group \(\mathbb{Z}_2\) are qubit codes.
- Spin code — Spin codes with spin \(\ell=1/2\) correspond to qubit codes.

## Zoo code information

## References

- [1]
- E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”. quant-ph/9608048
- [2]
- Upendra S. Kapshikar, “The Diagonal Distance of CWS Codes”. 2107.11286
- [3]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996). DOI; quant-ph/9604024
- [4]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001). DOI; cond-mat/0010440
- [5]
- Victor V. Albert, private communication, 2016
- [6]
- Steven M. Girvin, “Introduction to Quantum Error Correction and Fault Tolerance”. 2111.08894

## Cite as:

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