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arXiv:1902.07714 · Physical Review X

Continuous Symmetries and Approximate Quantum Error Correction

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

Published Oct 26, 2020• 107 citations
arXivHTMLPDF

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 nor 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.

DOI
10.1103/physrevx.10.041018

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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 nor 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.
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). 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. 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. 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. Explicit lower bounds are derived for both erasure and depolarizing noises. We also present a type of covariant codes which nearly saturates these lower bounds.
Error correcting codes with a universal set of transversal gates are a desideratum for quantum computing. Such codes, however, are ruled out by the Eastin-Knill theorem. Moreover, the theorem also rules out codes which are covariant with respect to the action of transversal unitary operations forming continuous symmetries. In this work, starting from an arbitrary code, we construct approximate codes which are covariant with respect to the entire group of local unitary gates in dimension $d\phantom{\rule{4pt}{0ex}}(<\ensuremath{\infty})$, using quantum reference frames. We show that our codes are capable of efficiently correcting different types of erasure errors. When only a small fraction of the nnn qudits upon which the code is built are erased, our covariant code has an error that scales as 1/n21/{n}^{2}1/n2, which is reminiscent of the Heisenberg limit of quantum metrology. When every qudit has a chance of being erased, our covariant code has an error that scales as 1/n1/n1/n. We show that the error scaling is optimal in both cases. Our approach has implications for fault-tolerant quantum computing, reference frame error correction, and the AdS-CFT duality.
Quantum error correction compatible with continuous symmetries is a fundamental problem in quantum information and a possible route to robust analog quantum simulation. Because the Eastin-Knill theorem forbids exact codes with continuous transversal symmetries, we construct explicit SU(d)SU(d)SU(d)-covariant approximate codes that exploit permutation symmetry to spread logical information uniformly across all physical subsystems. For one-, two-, and three-qudit erasures at known locations, we prove worst-case purified-distance scaling O(1/N)O(1/N)O(1/N), which matches known approximate Eastin-Knill lower bounds up to constants, and we extend the reduced-state analysis to general flagged local noise. Exact permutation symmetry of this kind is available only up to three erased sites; for an arbitrary fixed number kkk of erased sites, we instead show that Haar-random and efficiently sampled encodings achieve worst-case distance O(k/N)O(\sqrt{k}/N)O(k​/N) with probability exponentially close to one. For single-qudit erasure, we construct an explicit near-optimal decoder from the Petz recovery map. We then use these codes as building blocks for encoded analog dynamics. Symmetry-preserving Hamiltonians generate block-structured dynamical Lie algebras implementable transversally, while controlled symmetry-breaking terms serve as non-transversal resources for universal dynamics. These results provide explicit non-Abelian covariant codes and a framework for robust analog quantum simulation.
We systematically study the fundamental competition between quantum error correction (QEC) and continuous symmetries, two key notions in quantum information and physics, in a quantitative manner. Three meaningful measures of approximate symmetries in quantum channels and in particular QEC codes, respectively based on the violation of covariance conditions over the entire symmetry group or at a local point, and the violation of charge conservation, are introduced and studied. Each measure induces a corresponding characterization of approximately covariant codes. We explicate a host of different ideas and techniques that enable us to derive various forms of trade-off relations between the QEC inaccuracy and all symmetry violation measures. More specifically, we introduce two frameworks for understanding and establishing the trade-offs respectively based on the notions of charge fluctuation and gate implementation error, and employ methods including the Knill--Laflamme conditions as well as quantum metrology and quantum resource theory for the derivation. From the perspective of fault-tolerant quantum computing, our bounds on symmetry violation indicate limitations on the precision or density of transversally implementable logical gates for general QEC codes, refining the Eastin--Knill theorem. To exemplify nontrivial approximately covariant codes and understand the achievability of the above fundamental limits, we analyze the behaviors of two explicit types of codes: a parametrized extension of the thermodynamic code (which gives a construction of a code family that continuously interpolates between exact QEC and exact symmetry), and the quantum Reed--Muller codes. We show that both codes can saturate the scaling of the bounds for group-global covariance and charge conservation asymptotically, indicating the near-optimality of these bounds and codes.
We introduce an intrinsic formulation of quantum error correction based on representation theory, in which error-protection structure is encoded directly in a unitary group representation, rather than being tied to a particular embedding into a larger Hilbert space. In this framework, error models are classified according to the isotypic decomposition of the conjugation action on the operator algebra. Our main result, the \emph{Schur bootstrap}, shows that if an intrinsic code satisfies the Knill--Laflamme conditions on a given symmetry sector, then the same error-protection relations hold for every extrinsic realization obtained from a group-equivariant isometric embedding into a larger Hilbert space. Thus a single intrinsic verification certifies the corresponding symmetry-resolved error-correction conditions across an entire family of physical realizations. We further introduce an intrinsic notion of distance, called depth, defined via adjoint order. For standard multi-qudit systems this coincides with conventional code distance, while for more general representations it refines the usual weight-based notion. We also prove an intrinsic Eastin--Knill theorem: any intrinsic code of depth at least two has a discrete logical symmetry group, with the obstruction to continuous covariant gates arising from the representation-theoretic structure of the adjoint action. We illustrate the framework with several examples, including a minimal SU(2)\mathrm{SU}(2)SU(2) construction that unifies permutation-invariant qubit codes and bosonic codes, and higher-dimensional constructions exhibiting transversal Clifford symmetries and realizations beyond qubit systems.
Abstract Quantum error correction (QEC) is a key concept in quantum computation as well as many areas of physics. There are fundamental tensions between continuous symmetries and QEC. One vital situation is unfolded by the Eastin–Knill theorem, which forbids the existence of QEC codes that admit transversal continuous symmetry actions (transformations). Here, we systematically study the competition between continuous symmetries and QEC in a quantitative manner. We first define a series of meaningful measures of approximate symmetries motivated from different perspectives, and then establish a series of trade-off bounds between them and QEC accuracy utilizing multiple different methods. Remarkably, the results allow us to derive general quantitative limitations of transversally implementable logical gates, an important topic in fault-tolerant quantum computation. As concrete examples, we showcase two explicit types of quantum codes, obtained from quantum Reed–Muller codes and thermodynamic codes, respectively, that nearly saturate our bounds. Finally, we discuss several potential applications of our results in physics.