# Finite-dimensional quantum error-correcting code

## Description

## Protection

Denoting Hilbert spaces by the letter \(\mathsf{H}\), a finite-dimensional quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) with a set of correctable errors \({\cal{E}}:\mathsf{H}_N\to\mathsf{H}_M\) with the following property: there exists a quantum operation \({\cal{D}}:\mathsf{H}_M\to \mathsf{H}_K\) such that for all \(E\in\cal{E}\) and states \(|\psi\rangle\in\mathsf{H}_{K}\), \begin{align} {\cal D}(EU|\psi\rangle\langle\psi|U^{\dagger}E^{\dagger})=c(E,|\psi\rangle)|\psi\rangle\langle\psi|\tag*{(1)}\end{align} for some constant \(c\) [1]. A code is said to protect against or correct the errors \(\mathcal{E}\).

### Knill-Laflamme error correction conditions

Equivalently, correction capability is determined by of the quantum error-correction conditions, which may admit infinite terms due to non-normalizability of ideal code states. A code that satisfies these conditions approximately, i.e., up to some small quantifiable error, is called an approximate code.

Knill-Laflamme conditions: In a finite-dimensional Hilbert space, there are necessary and sufficient conditions for a code to successfully correct a set of errors. These are called the Knill-Laflamme error-correction conditions [2][3][4][5; Thm. 10.1]. A code defined by a partial isometry \(U\) with code space projector \(\Pi = U U^\dagger\) can correct a set of errors \(\{ E_j \}\) if and only if \begin{align} \Pi E_i^\dagger E_j \Pi = c_{ij}\, \Pi\qquad\text{for all \(i,j\),} \tag*{(2)}\end{align} where \(c_{ij}\) can be arbitrary numbers.

A code is degenerate with respect to a noise model if different errors map code states to the same error subspace. For a linearly independent error set \(\cal{E}\), degeneracy is equivalent to \(\text{rank}(c_{ij}) < |\cal{E}|\).

## Gates

## Decoding

## Parent

## Children

- Category-based quantum code
- Error-corrected sensing code — Semidefinite-program optimization procedure for finding a metrologically optimal code holds for finite-dimensional spaces.
- Quantum maximum-distance-separable (MDS) code
- Perfect quantum code
- Modular-qudit code
- Galois-qudit code
- Spin code

## Cousin

## References

- [1]
- D. Gottesman. Surviving as a quantum computer in a classical world
- [2]
- E. Knill and R. Laflamme, “Theory of quantum error-correcting codes”, Physical Review A 55, 900 (1997) DOI
- [3]
- C. H. Bennett et al., “Mixed-state entanglement and quantum error correction”, Physical Review A 54, 3824 (1996) arXiv:quant-ph/9604024 DOI
- [4]
- J. Preskill. Lecture notes on Quantum Computation. (1997–2020) URL
- [5]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2012) DOI
- [6]
- B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009) arXiv:0811.4262 DOI
- [7]
- I. Cong, S. Choi, and M. D. Lukin, “Quantum convolutional neural networks”, Nature Physics 15, 1273 (2019) arXiv:1810.03787 DOI
- [8]
- D. F. Locher, L. Cardarelli, and M. Müller, “Quantum Error Correction with Quantum Autoencoders”, (2023) arXiv:2202.00555

## Page edit log

- Victor V. Albert (2022-07-15) — most recent
- Victor V. Albert (2022-03-18)
- Victor V. Albert (2021-12-09)

## Cite as:

“Finite-dimensional quantum error-correcting code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qecc_finite

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/properties/qecc_finite.yml.