# Qubit code

## Description

## Protection

Corrects erasure errors on up to \(d-1\) qubits. The number of correctable errors is often called the decoding radius, and it is upper bounded by half of the code distance. As a result, qubit codes cannot tolerate adversarial errors on more than \((1-R)/4\) registers.

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}~. \tag*{(1)}\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][2][3]. 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 [4]. Codes whose distance is greater than the diagonal distance are degenerate.

## Decoding

## Fault Tolerance

## Notes

## Parents

- Modular-qudit code — Modular-qudit quantum codes for \(q=2\) correspond to codes.
- Galois-qudit code — Galois-qudit quantum codes for \(q=2\) correspond to qubit codes.
- Spin code — Spin codes with spin \(\ell=1/2\) correspond to qubit codes.

## Children

- Fermionic code — The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via the Jordan-Wigner transformation [8]. However, Majorana codes and the noise they are designed for are based on a different notion of locality.
- Codeword stabilized (CWS) code
- Movassagh-Ouyang Hamiltonian code
- XP stabilizer code

## Cousins

- Qubit c-q code — Qubit c-q codes are qubit codes designed to transmit classical information.
- 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 [9]. However, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [10].

## References

- [1]
- E. Knill, “Non-binary Unitary Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608048
- [2]
- E. Knill, “Group Representations, Error Bases and Quantum Codes”, (1996) arXiv:quant-ph/9608049
- [3]
- A. Klappenecker and M. Roetteler, “Beyond Stabilizer Codes I: Nice Error Bases”, (2001) arXiv:quant-ph/0010082
- [4]
- U. S. Kapshikar, “The Diagonal Distance of CWS Codes”, (2021) arXiv:2107.11286
- [5]
- O. Fawzi, A. Müller-Hermes, and A. Shayeghi, “A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation”, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022) arXiv:2202.00119 DOI
- [6]
- N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
- [7]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [8]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [9]
- Victor V. Albert, private communication, 2016
- [10]
- S. M. Girvin, “Introduction to Quantum Error Correction and Fault Tolerance”, (2023) arXiv:2111.08894

## Page edit log

- Victor V. Albert (2023-01-08) — most recent
- Sam Gunn (2022-01-08)
- Victor V. Albert (2022-05-07)
- Victor V. Albert (2021-10-29)

## Cite as:

“Qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubits_into_qubits