![]() Cappellaro, Quantum sensing, Reviews of Modern Physics 89, 035002 (2017). Maccone, Advances in quantum metrology, Nature Photonics 5, 222 (2011). Pastawski, Approximate recovery with locality and symmetry constraints, (2018), arXiv:1806.10324. Yard, A decoupling approach to the quantum capacity, Open Systems & Information Dynamics 15, 7 (2008). Oreshkov, General conditions for approximate quantum error correction and near-optimal recovery channels, Physical Review Letters 104, 120501 (2010). Bowen, Quantum error correcting codes in eigenstates of translation-invariant spin chains, Physical Review Letters 123, 110502 (2019). Tang, Quantum error-detection at low energies, Journal of High Energy Physics 2019, 21 (2019). ![]() Cubitt, Toy models of holographic duality between local hamiltonians, Journal of High Energy Physics 2019, 17 (2019). Ooguri, Symmetries in quantum field theory and quantum gravity, (2018), arXiv:1810.05338. Ooguri, Constraints on symmetries from holography, Physical Review Letters 122, 191601 (2019). Preskill, Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence, Journal of High Energy Physics 2015, 149 (2015). Harlow, Bulk locality and quantum error correction in ads/cft, Journal of High Energy Physics 2015, 163 (2015). Alhambra, Continuous groups of transversal gates for quantum error correcting codes from finite clock reference frames, Quantum 4, 245 (2020). Preskill, Quantum clock synchronization and quantum error correction, (2000), arXiv:quant-ph/0010098. Preskill, Continuous symmetries and approximate quantum error correction, Physical Review X 10 (2020). Salton, Error correction of quantum reference frame information, PRX Quantum 2 (2021). Laflamme, Quasi-exact quantum computation, Physical Review Research 2 (2020). Yoder, Disjointness of stabilizer codes and limitations on fault-tolerant logical gates, Physical Review X 8, 021047 (2018). Yoshida, Fault-tolerant logical gates in quantum error-correcting codes, Physical Review A 91, 012305 (2015). König, Classification of topologically protected gates for local stabilizer codes, Physical Review Letters 110, 170503 (2013). Knill, Restrictions on transversal encoded quantum gate sets, Physical Review Letters 102, 110502 (2009). Brun, Quantum error correction (Cambridge university press, 2013). Gottesman, in Quantum information science and its contributions to mathematics, Proceedings of Symposia in Applied Mathematics, Vol. ![]() Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010). We also present a type of covariant codes which nearly saturates these lower bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. We prove new and powerful lower bounds on the infidelity of covariant quantum error correction, which not only extend the scope of previous no-go results but also provide a substantial improvement over existing bounds. Here, we explore covariant quantum error correction with respect to continuous symmetries from the perspectives of quantum metrology and quantum resource theory, establishing solid connections between these formerly disparate fields. The need for understanding the limits of covariant quantum error correction arises in various realms of physics including fault-tolerant quantum computation, condensed matter physics and quantum gravity. Covariant codes are quantum codes such that a symmetry transformation on the logical system could be realized by a symmetry transformation on the physical system, usually with limited capability of performing quantum error correction (an important case being the Eastin–Knill theorem). ![]()
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