Continuous Symmetries and Approximate Quantum Error Correction

Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encod...

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Autores principales:	Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, John Preskill
Formato:	article
Lenguaje:	EN
Publicado:	American Physical Society 2020
Materias:	Physics QC1-999
Acceso en línea:	https://doaj.org/article/0bade365038648be8d78e2a2c3b1e0fe
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spelling	oai:doaj.org-article:0bade365038648be8d78e2a2c3b1e0fe2021-12-02T11:17:46ZContinuous Symmetries and Approximate Quantum Error Correction10.1103/PhysRevX.10.0410182160-3308https://doaj.org/article/0bade365038648be8d78e2a2c3b1e0fe2020-10-01T00:00:00Zhttp://doi.org/10.1103/PhysRevX.10.041018http://doi.org/10.1103/PhysRevX.10.041018https://doaj.org/toc/2160-3308Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.Philippe FaistSepehr NezamiVictor V. AlbertGrant SaltonFernando PastawskiPatrick HaydenJohn PreskillAmerican Physical SocietyarticlePhysicsQC1-999ENPhysical Review X, Vol 10, Iss 4, p 041018 (2020)
institution	DOAJ
collection	DOAJ
language	EN
topic	Physics QC1-999
spellingShingle	Physics QC1-999 Philippe Faist Sepehr Nezami Victor V. Albert Grant Salton Fernando Pastawski Patrick Hayden John Preskill Continuous Symmetries and Approximate Quantum Error Correction
description	Quantum error correction and symmetry arise in many areas of physics, including many-body systems, metrology in the presence of noise, fault-tolerant computation, and holographic quantum gravity. Here, we study the compatibility of these two important principles. If a logical quantum system is encoded into n physical subsystems, we say that the code is covariant with respect to a symmetry group G if a G transformation on the logical system can be realized by performing transformations on the individual subsystems. For a G-covariant code with G a continuous group, we derive a lower bound on the error-correction infidelity following erasure of a subsystem. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. We exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. Leveraging tools from representation theory, we prove an approximate version of the Eastin-Knill theorem for quantum computation: If a code admits a universal set of transversal gates and corrects erasure with fixed accuracy, then, for each logical qubit, we need a number of physical qubits per subsystem that is inversely proportional to the error parameter. We construct codes covariant with respect to the full logical unitary group, achieving good accuracy for large d (using random codes) or n (using codes based on W states). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. In the context of the AdS/CFT correspondence, our approach provides insight into how time evolution in the bulk corresponds to time evolution on the boundary without violating the Eastin-Knill theorem, and our five-rotor code can be stacked to form a covariant holographic code.
format	article
author	Philippe Faist Sepehr Nezami Victor V. Albert Grant Salton Fernando Pastawski Patrick Hayden John Preskill
author_facet	Philippe Faist Sepehr Nezami Victor V. Albert Grant Salton Fernando Pastawski Patrick Hayden John Preskill
author_sort	Philippe Faist
title	Continuous Symmetries and Approximate Quantum Error Correction
title_short	Continuous Symmetries and Approximate Quantum Error Correction
title_full	Continuous Symmetries and Approximate Quantum Error Correction
title_fullStr	Continuous Symmetries and Approximate Quantum Error Correction
title_full_unstemmed	Continuous Symmetries and Approximate Quantum Error Correction
title_sort	continuous symmetries and approximate quantum error correction
publisher	American Physical Society
publishDate	2020
url	https://doaj.org/article/0bade365038648be8d78e2a2c3b1e0fe
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Continuous Symmetries and Approximate Quantum Error Correction

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