# Covariant code[1]

## Description

Code constructed in a physical space consisting of a tensor product of \(n\) subsystems (e.g., qubits, modular qudits, or Galois qudits) 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\,, \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 [5][6][7][3][4].

## Transversal Gates

## Parent

## Children

- \(G\)-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
- \([[3,1,2]]_{\mathbb Z}\) rotor code — The \([[3,1,2]]_{\mathbb Z}\) code is \(U(1)\)-covariant.
- \([[5,1,3]]_{\mathbb Z}\) rotor code — The \([[5,1,3]]_{\mathbb Z}\) code is \(U(1)\)-covariant.

## 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][3].
- Eigenstate thermalization hypothesis (ETH) code — ETH codes consisting of Dicke states are approximately \(U(1)\)-covariant and nearly saturate certain covariance-performance bounds [5][3][3].
- Random quantum code — Random \(U(d)\)-covariant almost exactly error-correcting codes exist [5][8].
- 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.

## References

- [1]
- P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021). DOI; 1709.04471
- [2]
- Philippe Faist et al., “Time-energy uncertainty relation for noisy quantum metrology”. 2207.13707
- [3]
- Zi-Wen Liu and Sisi Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”. 2111.06360
- [4]
- Zi-Wen Liu and Sisi Zhou, “Approximate symmetries and quantum error correction”. 2111.06355
- [5]
- P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020). DOI; 1902.07714
- [6]
- S. Zhou, Z.-W. Liu, and L. Jiang, “New perspectives on covariant quantum error correction”, Quantum 5, 521 (2021). DOI; 2005.11918
- [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). DOI; 2004.11893
- [8]
- L. Kong and Z.-W. Liu, “Near-Optimal Covariant Quantum Error-Correcting Codes from Random Unitaries with Symmetries”, PRX Quantum 3, (2022). DOI; 2112.01498

## Zoo code information

## Cite as:

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