## Description

## Protection

Denoting Hilbert spaces by the letter \(\mathsf{H}\), a finite-dimensional quantum code \((U,\cal{E})\) is a partial isometry \(U:\mathsf{H}_{K}\to\mathsf{H}_{N}\) and 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: The Knill-Laflamme error-correction conditions [2–4][5; Thm. 10.1] are necessary and sufficient conditions for a code to successfully correct a set of errors in a finite-dimensional Hilbert space. 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 the QEC matrix elements \(c_{ij}\) are arbitrary complex numbers.

The Knill-Laflamme conditions can alternatively be expressed in terms of the complementary channel, or in an information-theoretic way via a data processing inequality [6,7][8; Eq. (29)]. They motivate higher-rank numerical ranges, which are generalizations of the numerical range of an operator [9,10]. They have been extended to sequences of multiple errors and rounds of correction [11].

Degeneracy: 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 \(\text{rank}(c_{ij}) < |\cal{E}|\) [1].

### Correctability of quantum channels

From now on, we use \(\mathcal{E}\) to denote a noise channel constructed out of the set of errors \(E\) and let \(\mathcal{U}(\cdot)=U(\cdot)U^\dagger\) be the superoperator corresponding to the partial encdoing isometry \(U\). A noise channel is correctable if there exists a recovery channel \(\mathcal{D}\) such that \begin{align} \mathcal{D}\mathcal{E}\mathcal{U}(\rho)=\rho \tag*{(3)}\end{align} for all logical states \(\rho\).

The above is equivalent to the fidelity between \(\rho\) and \(\mathcal{D}\mathcal{E}\mathcal{U}(\rho)\) being one for any notion of distance between quantum states. In particular, we can consider a scenario where we send only one part of an entangled state through a channel and determine whether the entanglement has been preserved during transmission. Using the notion of entanglement fidelity, a quantum channel \(\mathcal{E}\) is exactly correctable iff there exists a quantum channel \(\mathcal{D}\) such that \begin{align} (\mathcal{D}\mathcal{E}\mathcal{U}\otimes\mathrm{id})(\ket{\psi}\bra{\psi})=\ket{\psi}\bra{\psi} \tag*{(4)}\end{align} for all states \(\rho\) and their corresponding purifications \(\ket{\psi}\) (i.e., states \(\ket{\psi}\) such that \(\text{Tr}_{2}(|\psi\rangle\langle\psi|)=\rho\)).

The above entanglement fidelity condition can be alternatively expressed using complementary channels.

Complementary channel: A complementary channel \(\mathcal{E}^C\) is obtained from a channel \(\mathcal{E}\) that acts on a system by interpreting the channel as coming from a unitary operation acting on a larger system-environment tensor-product space (i.e., performing an isometric extension) and then tracing out the system factor (instead of the second environmental factor) [12; Sec. 5.2.2]. A noise channel \({\cal E}(\cdot)=\sum_{j}E_{j}(\cdot)E_{j}^{\dagger}\) admits a complementary channel of the form \begin{align} {\cal E}^{C}(\cdot)=\sum_{j,k}\text{Tr}\{E_{j}(\cdot)E_{k}^{\dagger}\}|j\rangle\langle k|~. \tag*{(5)}\end{align}

A channel \(\mathcal{E}\) is correctable if \(\mathcal{E}^C(\rho)=\rho_0\mathrm{Tr}(\rho)\) for some constant state \(\rho_0\), which is equivalent to the Knill-Laflamme conditions [13,14].

## Rate

## Decoding

## Code Capacity Threshold

## 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
- Spin code

## Cousin

## References

- [1]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [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. Schumacher and M. A. Nielsen, “Quantum data processing and error correction”, Physical Review A 54, 2629 (1996) arXiv:quant-ph/9604022 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. 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
- [9]
- M.-D. Choi, D. W. Kribs, and K. Życzkowski, “Quantum error correcting codes from the compression formalism”, Reports on Mathematical Physics 58, 77 (2006) arXiv:quant-ph/0511101 DOI
- [10]
- M.-D. Choi, D. W. Kribs, and K. Zyczkowski, “Higher-Rank Numerical Ranges and Compression Problems”, (2006) arXiv:math/0511278
- [11]
- A. Tanggara, M. Gu, and K. Bharti, “Strategic Code: A Unified Spatio-Temporal Framework for Quantum Error-Correction”, (2024) arXiv:2405.17567
- [12]
- M. M. Wilde, Quantum Information Theory (Cambridge University Press, 2013) DOI
- [13]
- D. W. Kribs, A. Pasieka, and K. Zyczkowski, “Entropy of a quantum error correction code”, (2008) arXiv:0811.1621
- [14]
- C. Bény and O. Oreshkov, “General Conditions for Approximate Quantum Error Correction and Near-Optimal Recovery Channels”, Physical Review Letters 104, (2010) arXiv:0907.5391 DOI
- [15]
- J. Watrous, The Theory of Quantum Information (Cambridge University Press, 2018) DOI
- [16]
- J. P. Paz and W. H. Zurek, “Continuous error correction”, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 454, 355 (1998) DOI
- [17]
- J. P. Barnes and W. S. Warren, “Automatic Quantum Error Correction”, Physical Review Letters 85, 856 (2000) arXiv:quant-ph/9912104 DOI
- [18]
- C. Ahn, A. C. Doherty, and A. J. Landahl, “Continuous quantum error correction via quantum feedback control”, Physical Review A 65, (2002) arXiv:quant-ph/0110111 DOI
- [19]
- M. Sarovar and G. J. Milburn, “Continuous quantum error correction by cooling”, Physical Review A 72, (2005) arXiv:quant-ph/0501038 DOI
- [20]
- R. van Handel and H. Mabuchi, “Optimal error tracking via quantum coding and continuous syndrome measurement”, (2005) arXiv:quant-ph/0511221
- [21]
- J. F. Poyatos, J. I. Cirac, and P. Zoller, “Quantum Reservoir Engineering with Laser Cooled Trapped Ions”, Physical Review Letters 77, 4728 (1996) DOI
- [22]
- O. Oreshkov, “Continuous-time quantum error correction”, (2013) arXiv:1311.2485
- [23]
- J.-M. Lihm, K. Noh, and U. R. Fischer, “Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation”, Physical Review A 98, (2018) arXiv:1711.02999 DOI
- [24]
- O. Oreshkov and T. A. Brun, “Continuous quantum error correction for non-Markovian decoherence”, Physical Review A 76, (2007) arXiv:0705.2342 DOI
- [25]
- L. Colmenarez et al., “Accurate optimal quantum error correction thresholds from coherent information”, (2024) arXiv:2312.06664

## Page edit log

- Victor V. Albert (2023-07-24) — most recent
- Milan Tenn (2023-07-17)
- Milan Tenn (2023-07-07)
- Milan Tenn (2023-07-03)
- Milan Tenn (2023-06-28)
- Victor V. Albert (2022-07-15)
- 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.), 2023. https://errorcorrectionzoo.org/c/qecc_finite

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