Continuous Symmetries and Approximate Quantum Error Correction

Abstract

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.

• Received 12 April 2019
• Revised 27 July 2020
• Accepted 7 September 2020

DOI: https://doi.org/10.1103/PhysRevX.10.041018

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

1. Research Areas

1. Properties

1. Techniques

Quantum Information Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Philippe Faist^1,2,*,†, Sepehr Nezami^3,†, Victor V. Albert ^1,4, Grant Salton^1,3, Fernando Pastawski², Patrick Hayden³, and John Preskill^1,4

• ¹Institute for Quantum Information and Matter, Caltech, Pasadena, 91125 California, USA
• ²Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
• ³Stanford Institute for Theoretical Physics, Stanford University, Stanford, 94305 California, USA
• ⁴Walter Burke Institute for Theoretical Physics, Caltech, Pasadena, 91125 California, USA

• ^*Corresponding author. philippe.faist@fu-berlin.de
• ^†These authors contributed equally to this work.

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Popular Summary

Quantum error correction (QEC) enables a quantum computer to operate reliably even when the computing hardware is prone to error. In addition to its importance for future quantum technologies, QEC is also a fundamental principle of physics, with far-ranging implications for fields as diverse as quantum chaos, topological quantum states of matter, and quantum gravity. Here, we study a fundamental tension between the continuous symmetries of a QEC code and the code's effectiveness.

Any message—including quantum states—can be protected by redundancies. A quantum error-correcting code records a quantum state using a number of "letters," where each letter is a multidimensional quantum system, such that the state can be recovered even if some of the letters are destroyed. Such a code is said to be covariant with respect to a symmetry if the encoded state can be transformed by applying that transformation to each of the code's letters. Previous work had shown that no code covariant with respect to a continuous symmetry can provide perfect protection against an error in which one of the letters is lost. We show that loss of a letter can be approximately corrected by a covariant code, with a residual error that becomes arbitrarily small when either the number of letters or dimensions is large.

Our results clarify how quantum states can robustly convey information about a reference frame. Covariant codes are also of potential interest in quantum computing, because protected information can be processed by acting on the individual letters rather than collectively on several letters, though there are serious limitations on the reliability of such schemes for processing information.

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Article Text

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References

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Source: https://link.aps.org/doi/10.1103/PhysRevX.10.041018

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