## 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 [1–3]. 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.

A quantum channel whose Kraus operators are Pauli strings is called a Pauli channel, and such channels are typically more tractable than general, non-Pauli channels. Relevant Pauli channels include dephasing noise and depolarizing noise (a.k.a. Werner-Holevo channel [5]), while relevant non-Pauli channels are amplitude damping, erasure (which maps all qubit states into a third state \(|e\rangle\) outside of the qubit Hilbert space), and biased erasure (in which case only the \(|1\rangle\) qubit state is mapped to \(|e\rangle\)).

Bounds on code performance include the quantum Singleton bound [6–8], quantum Hamming bound [9], and quantum Gilbert-Varshamov bound [9]. Linear programming bounds also exist [10,11].

Determining protection and bounds on code parameters can also be done using Shor-Laflamme quantum weight enumerators [12] and Rains shadow enumerators [10] (see also [13]).

## Rate

## Gates

## Decoding

## Fault Tolerance

## Threshold

Measurement threshold: One can derive conditions quantifying how many random single-qubit measurements can be made without destroying the logical information [20]. The measurement threshold is the maximum total probability that a single qubit is measured in a random \(X\), \(Y\), or \(Z\) basis at which the logical information is still recoverable. The measurement threshold is at least as large as the erasure threshold [20; Thm. 4].

## Notes

## Parents

- Modular-qudit code — Modular-qudit quantum codes for \(q=2\) correspond to qubit 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

- Haar-random qubit code
- Local Haar-random circuit qubit code
- Fermionic code — The Majorana operator algebra is isomorphic to the qubit Pauli-operator algebra via the Jordan-Wigner transformation [22]. However, Majorana codes and the noise they are designed for are based on a different notion of locality.
- Eigenstate thermalization hypothesis (ETH) code
- Movassagh-Ouyang Hamiltonian code
- Matrix-product state (MPS) code
- Quotient space quantum code (QSQC)
- XP stabilizer code
- Neural network 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 [23]. However, noise models for the two code families induce different notions of locality and thus qualitatively different physical interpretations [24].
- EA qubit code — EA qubit codes utilize additional ancillary qubits in a pre-shared entangled state, but reduce to ordinary qubit codes when said qubits are interpreted as noiseless physical qubits.
- Five-qubit perfect code — Every \(((5,2,3))\) code is equivalent to the five-qubit code [25; Corr. 10].
- Subsystem qubit code — Subsystem qubit codes reduce to (subspace) qubit codes when there is no gauge subsystem.
- Single-spin code — Certain single-spin codes yield permutation-invariant qubit codes with non-trivial distance [26] when the single spin is treated as a collective spin of several qubits.

## 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]
- R. F. Werner and A. S. Holevo, “Counterexample to an additivity conjecture for output purity of quantum channels”, Journal of Mathematical Physics 43, 4353 (2002) arXiv:quant-ph/0203003 DOI
- [6]
- E. Knill, R. Laflamme, and L. Viola, “Theory of Quantum Error Correction for General Noise”, Physical Review Letters 84, 2525 (2000) arXiv:quant-ph/9604034 DOI
- [7]
- N. J. Cerf and R. Cleve, “Information-theoretic interpretation of quantum error-correcting codes”, Physical Review A 56, 1721 (1997) arXiv:quant-ph/9702031 DOI
- [8]
- E. M. Rains, “Nonbinary quantum codes”, (1997) arXiv:quant-ph/9703048
- [9]
- A. Ekert and C. Macchiavello, “Error Correction in Quantum Communication”, (1996) arXiv:quant-ph/9602022
- [10]
- E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
- [11]
- A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
- [12]
- P. Shor and R. Laflamme, “Quantum MacWilliams Identities”, (1996) arXiv:quant-ph/9610040
- [13]
- A. J. Scott, “Probabilities of Failure for Quantum Error Correction”, Quantum Information Processing 4, 399 (2005) arXiv:quant-ph/0406063 DOI
- [14]
- S. Pirandola et al., “Fundamental limits of repeaterless quantum communications”, Nature Communications 8, (2017) arXiv:1510.08863 DOI
- [15]
- A. Barenco et al., “Elementary gates for quantum computation”, Physical Review A 52, 3457 (1995) arXiv:quant-ph/9503016 DOI
- [16]
- J. van de Wetering and M. Amy, “Optimising T-count is NP-hard”, (2023) arXiv:2310.05958
- [17]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [18]
- O. Fawzi, A. Müller-Hermes, and A. Shayeghi, “A Lower Bound on the Space Overhead of Fault-Tolerant Quantum Computation”, (2022) arXiv:2202.00119 DOI
- [19]
- N. Baspin, O. Fawzi, and A. Shayeghi, “A lower bound on the overhead of quantum error correction in low dimensions”, (2023) arXiv:2302.04317
- [20]
- D. Lee and B. Yoshida, “Randomly Monitored Quantum Codes”, (2024) arXiv:2402.00145
- [21]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [22]
- A. Y. Kitaev, “Unpaired Majorana fermions in quantum wires”, Physics-Uspekhi 44, 131 (2001) arXiv:cond-mat/0010440 DOI
- [23]
- Victor V. Albert, private communication, 2016
- [24]
- S. M. Girvin, “Introduction to quantum error correction and fault tolerance”, SciPost Physics Lecture Notes (2023) arXiv:2111.08894 DOI
- [25]
- E. M. Rains, “Quantum codes of minimum distance two”, (1997) arXiv:quant-ph/9704043
- [26]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI

## 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