Continuous Symmetries and Approximate Quantum Error Correction

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Continuous Symmetries and Approximate Quantum Error Correction

Philippe Faist, Sepehr Nezami, Victor V. Albert, Grant Salton, Fernando Pastawski, Patrick Hayden, and John Preskill

Phys. Rev. X 10, 041018 – Published 26 October 2020

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

Physics Subject Headings (PhySH)

Quantum channels Quantum error correction Quantum gates Quantum information processing Quantum metrology

Symmetries

O(N) symmetry SU(N) symmetries U(N) symmetry

Quantum Information, Science & TechnologyCondensed 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.

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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Vol. 10, Iss. 4 — October - December 2020

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Figure 1
Quantum information represented on an abstract logical system $L$ is encoded on several physical subsystems $A_{1} \dots A_{n}$ using a code. Suppose that the code is compatible with a continuous transversal symmetry, for instance, rotations in 3D space. This compatibility means that by rotating all individual physical subsystems one induces the same transformation as if one had simply rotated the initial logical system $L$ . No such code provides exact protection against erasure of a single subsystem. To understand why, consider two eigenstates with different eigenvalues of the logical charge $T_{L}$ that generates the symmetry. Transversality means that the corresponding physical charge $T_{A} = \sum T_{i}$ is a sum of terms, each supported on a single subsystem. By measuring the charge $T_{i}$ of a random subsystem, the environment can infer information about the total charge $T_{L}$ , distinguishing the two eigenstates. It follows that erasure of a single subsystem cannot be perfectly corrected.
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Figure 2
Overview of our main results and structure of the paper.
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Figure 3
A code $E_{L \to A}$ maps a logical state $| x ⟩$ on an abstract logical space $L$ to a state $| ψ_{x} ⟩_{A}$ on a physical system $A$ . Here, we consider a physical system composed of several subsystems $A = A_{1} \otimes A_{2} \otimes \dots A_{n}$ . The code space, with associated projector $Π_{A}$ , is the range of the encoding map. The environment acts by erasing a subsystem $A_{i}$ , represented as a noise channel $N_{A \to A}^{(i)}$ . A good error-correcting code is capable of recovering the original logical state $| x ⟩$ from the remaining subsystems, by applying a recovery map $R_{A \to L}^{(i)}$ . In our analysis, we assume that the environment chooses randomly which subsystem $A_{i}$ is erased and that $i$ is known, so that the recovery map may depend on $i$ . The quality of the code is characterized by how close the overall process (erasure followed by recovery) is to the identity process acting on the logical system, as measured by either the (fixed-input) entanglement fidelity or the worst-case entanglement fidelity.
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Figure 4
The general setting of Theorem 2. An approximately charge-conserving encoding maps a logical state onto several physical systems. The noise acts by randomly erasing some subsystems and storing a record of which systems are erased in a register $C$ . Which combinations of subsystems can be lost and with which probability can be chosen arbitrarily. The continuous symmetry is assumed to have a generator represented by $T_{L}$ on the logical system and by $T_{A}$ on the physical systems. We assume that $T_{A}$ can be written as a sum of terms $T_{A} = \sum T_{α}$ , where each $T_{α}$ acts on a combination of subsystems that could possibly be lost to the environment. For instance, $T_{A}$ may include a term $T_{3, 4, 7}$ acting on systems $A_{3} A_{4} A_{7}$ only if the noise model is such that the systems 3, 4, and 7 have a nonzero probability of being simultaneously erased. The different terms may overlap and do not have to commute.
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Figure 5
Smoothness of the Littlewood-Richardson coefficients, required for our proof that random covariant codes can asymptotically correct against errors. A Littlewood-Richardson coefficient $c_{μ ν}^{λ}$ is the coefficient that counts the degeneracy of the $U (d_{L})$ irrep labeled by the Young diagram $λ$ in the tensor product of two other irreps labeled by $μ$ and $ν$ . The Littlewood-Richardson coefficients are nonzero in a convex cone in the space of $(μ, ν, λ)$ . This cone—or chamber complex—is divided into several smaller convex cones—or chambers—in which $c_{μ ν}^{λ}$ is a polynomial of $μ$ , $ν$ , and $λ$ . Hence, a generic choice of irreps on which we choose a random code has corresponding coefficients that are smooth, which we show implies good asymptotic performance of the code.
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Figure 6
Nontrivial bulk global symmetries are incompatible with the AdS/CFT quantum error-correcting code. A bulk global symmetry operator $U_{L} (g)$ corresponds to a boundary global symmetry operator $U_{CFT} (g)$ which is transversal with respect to the decomposition of the boundary into subregions ${A_{k}}$ : $U_{CFT} (g) = \underset{k}{\otimes} W_{k} (g)$ , where $W_{k} (g)$ is supported on $A_{k}$ . The bulk subregion $a_{0}$ is outside the entanglement wedge $a_{k}$ of each boundary subregion $A_{k}$ ; therefore, erasure of each $A_{k}$ is correctable for the algebra $A$ of bulk local operators on $a_{0}$ . Furthermore, each $W_{k} (g)$ maps low-energy states of the boundary CFT to low-energy states. This result means that $W_{k} (g)$ is a logical operator which preserves the code space of the CFT. A logical operator $W_{k} (g)$ supported on a correctable boundary subregion $A_{k}$ must be the logical identity. Therefore, the global symmetry operator $U_{L} (g)$ acts trivially on bulk local operators.
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