Covariant code[1] 


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].


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 [37].

Transversal Gates

\(G\)-covariant codes defined on a tensor product space consisting of \(n\) subsystems are equivalent to codes with a transversal gate set realizing \(G\).





P. Hayden et al., “Error Correction of Quantum Reference Frame Information”, PRX Quantum 2, (2021) arXiv:1709.04471 DOI
P. Faist et al., “Time-Energy Uncertainty Relation for Noisy Quantum Metrology”, PRX Quantum 4, (2023) arXiv:2207.13707 DOI
Z.-W. Liu and S. Zhou, “Quantum error correction meets continuous symmetries: fundamental trade-offs and case studies”, (2023) arXiv:2111.06360
Z.-W. Liu and S. Zhou, “Approximate symmetries and quantum error correction”, npj Quantum Information 9, (2023) arXiv:2111.06355 DOI
P. Faist et al., “Continuous Symmetries and Approximate Quantum Error Correction”, Physical Review X 10, (2020) arXiv:1902.07714 DOI
S. Zhou, Z.-W. Liu, and L. Jiang, “New perspectives on covariant quantum error correction”, Quantum 5, 521 (2021) arXiv:2005.11918 DOI
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
A. Denys and A. Leverrier, “Multimode bosonic cat codes with an easily implementable universal gate set”, (2023) arXiv:2306.11621
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
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
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Zoo Code ID: covariant

Cite as:
“Covariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_covariant, title={Covariant code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Covariant code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.