# Covariant code[1]

## Description

A block code on \(n\) subsystems that admits a group \(G\) of transversal gates. The group has to be finite for finite-dimensional codes due to the Eastin-Knill theorem. Continuous-\(G\) covariant codes, necessarily infinite-dimensional, are relevant to error correction of quantum reference frames [1] and error-corrected parameter estimation.

Denoting the code's encoding map as \(U\), covariance is equivalent to \begin{align} \left(\bigotimes_{j=1}^{n}V_{j}\left(g\right)\right)U=UV_{L}\left(g\right)\quad\quad\forall g\in G\,, \tag*{(1)}\end{align} where \(V_j(g)\) is a unitary representation of \(g\) acting on the \(j\) subsystem, and \(V_L\) is a unitary representation acting on the unencoded logical information. In this way, covariant encoding maps are equivariant (i.e., commute) with group actions on the logical and physical spaces.

Almost always, the physical representation is defined to be the transversal one (with respect to some tensor-product decomposition), but can reduce to any representation when the code is a subspace of a larger space that is not expressed as a tensor product (\(n=1\)). More generally, a code is sometimes said to be time-covariant if it admits a continuous-parameter \(U(1)\) family of gates, not necessarily transversal [2].

## Protection

Finite-dimensional codes correcting a single-subsystem erasure and admitting a continuous-parameter family of transversal gates (assuming \(n>1\)) cannot exist in finite dimensions due to the Eastin-Knill theorem. As a result, there is generally a tradeoff between covariance and error correction.

Exact error-correcting \(G\)-covariant codes can exist in infinite dimensions, but their codewords are non-normalizable, meaning that approximate constructions have to be considered that are only approximately error correcting. On the other hand, there exist exact error-correcting codes in finite dimensions that are approximately covariant [3,4]. Various bounds quantify the covariance-performance tradeoff [3–7].

## Transversal Gates

## Parents

- Block quantum code — Covariant codes for \(n>1\) are block quantum codes.
- Group-representation code — Covariant codes are block group-representation codes [8; Lemma 2].

## Children

- Three-rotor code — The three-rotor code is \(U(1)\)-covariant.
- Five-rotor code — The five-rotor code is \(U(1)\)-covariant.
- \([[4,2,2]]_{G}\) four group-qudit code — The four group-qudit code is \(G\)-covariant.
- \(G\)-covariant erasure code — In a proof of principle demonstration, error-correcting codes that are finite-\(G\) covariant can be constructed from a base encoding \(U_0\).
- \(U(d)\)-covariant approximate erasure code
- W-state code — The W-state code approximately protects against a single erasure while allowing for a universal transversal set of gates.

## Cousins

- Approximate quantum error-correcting code (AQECC) — Normalizable constructions of infinite-dimensional \(G\)-covariant codes for continuous \(G\) are approximately error-correcting.
- Quantum Reed-Muller code — Quantum RM codes are approximately covariant and nearly saturate certain covariance-performance bounds [3].
- Eigenstate thermalization hypothesis (ETH) code — ETH codes consisting of Dicke states are approximately \(U(1)\)-covariant and nearly saturate certain covariance-performance bounds [3,5].
- Random quantum code — Random \(U(d)\)-covariant almost exactly error-correcting codes exist [5,9].
- Group GKP code — Group-GKP codes corresponding to the \(G^{k_1} \subseteq G^{k_2} \subset G^{n}\) group construction admit \(X\)-type transversal Pauli gates that represent the group \(G\), and are thus \(G\)-covariant [5].
- Metrological code — Any time-covariant QECC, i.e., a code admitting a continuous-parameter \(U(1)\) family of gates, is automatically a metrological code.
- Valence-bond-solid (VBS) code — Two classes of (approximate) VBS codes have \(SU(q)\) transversal gates, i.e., are \(SU(q)\)-covariant [10; Tab. III].

## References

- [1]
- P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021) arXiv:1709.04471 DOI
- [2]
- P. Faist et al., “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI
- [3]
- Z.-W. Liu and S. Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”, (2023) arXiv:2111.06360
- [4]
- Z.-W. Liu and S. Zhou, “Approximate symmetries and quantum error correction”, npj Quantum Information 9, (2023) arXiv:2111.06355 DOI
- [5]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
- [6]
- S. Zhou, Z.-W. Liu, and L. Jiang, “New perspectives on covariant quantum error correction”, Quantum 5, 521 (2021) arXiv:2005.11918 DOI
- [7]
- A. Kubica and R. Demkowicz-Dobrzański, “Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem”, Physical Review Letters 126, (2021) arXiv:2004.11893 DOI
- [8]
- A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
- [9]
- L. Kong and Z.-W. Liu, “Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries”, PRX Quantum 3, (2022) arXiv:2112.01498 DOI
- [10]
- D.-S. Wang et al., “Theory of quasi-exact fault-tolerant quantum computing and valence-bond-solid codes”, New Journal of Physics 24, 023019 (2022) arXiv:2105.14777 DOI

## Page edit log

- Victor V. Albert (2022-07-30) — most recent
- Jack Davis (2022-04-02)

## Cite as:

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