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arXiv:2607.07607

Covariant Approximate Quantum Codes for Protected Analog Computation

Mariia Elovenkova, Hong-Ye Hu, Susanne F. Yelin

quant-phSubmitted Jul 8, 2026• Announced Jul 9, 2026• 0 citations
v1 · Jul 8, 2026v2 · Sep 1, 2026
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Abstract

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 $SU(d)$-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 $Θ(1/N)$, matching approximate Eastin-Knill lower bounds up to constants, and we extend the reduced-state analysis to general flagged local noise. 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.

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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 Θ(1/N)Θ(1/N)Θ(1/N), matching approximate Eastin-Knill lower bounds up to constants, and we extend the reduced-state analysis to general flagged local noise. 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.
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.
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 and symmetries play central roles in quantum information science and physics. It is known that quantum error-correcting codes covariant with respect to continuous symmetries cannot correct erasure errors perfectly (an important case being the Eastin-Knill theorem), in contrast to the case without symmetry constraints. Furthermore, there are fundamental limits on the accuracy of such covariant codes for approximate quantum error correction. Here, we consider the quantum error correction capability of random covariant codes. In particular, we show that U(1)U(1)U(1)-covariant codes generated by Haar random U(1)U(1)U(1)-symmetric unitaries, i.e. unitaries that commute with the charge operator (or conserve the charge), typically saturate the fundamental limits to leading order in terms of both the average- and worst-case purified distances against erasure noise. We note that the results hold for symmetric variants of unitary 2-designs, and comment on the convergence problem of charge-conserving random circuits. Our results not only indicate (potentially efficient) randomized constructions of optimal U(1)U(1)U(1)-covariant codes, but also reveal fundamental properties of random charge-conserving unitaries, which may underlie important models of complex quantum systems in wide-ranging physical scenarios where conservation laws are present, such as black holes and many-body spin systems.
Quantum information processing in the presence of continuous symmetry is of wide importance and exhibits many novel physical and mathematical phenomena. SU(d) is a continuous symmetry group of particular interest since it represents a fundamental type of non-Abelian symmetry and also plays a vital role in quantum computation. Here, we explicate three particularly interesting applications of SU(d)-symmetric random unitaries in diverse contexts ranging from physics to quantum computing: information scrambling with non-Abelian conserved quantities, covariant quantum error correcting random codes, and geometric quantum machine learning. First, we show that, in the presence of SU(d) symmetry, the local conserved quantities would exhibit residual values even at t → ∞ which decays as Ω(1/n3/2) under local Pauli basis for qubits and Ω(1/n(d+2)2/2)\Omega (1/{n}^{{(d+2)}^{2}/2})Ω(1/n(d+2)2/2) under local symmetric basis for general qudits with respect to the system size, in contrast to O(1/n) decay for U(1) case and the exponential decay for no-symmetry case in the sense of out-of-time ordered correlator (OTOC). Second, we show that SU(d)-symmetric unitaries can be used to construct asymptotically optimal (in the sense of saturating the fundamental limits on the code error, or the approximate Eastin–Knill theorems) SU(d)-covariant codes (codes with universal transversal gates) that encode any constant number of logical qudits, extending [Kong & Liu; PRXQ 3, 020314 (2022)]. Finally, we derive an overpartameterization threshold via the quantum neural tangent kernel (QNTK) required for exponential convergence guarantee of generic ansatz for geometric quantum machine learning, which reveals that the number of parameters required scales only with the dimension of desired subspaces rather than that of the entire Hilbert space. Our work invites further research on quantum information with continuous symmetries, where the mathematical tools developed in this work are expected to be useful.
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.
Fast preparation of quantum error-correcting codes is essential for scalable quantum memories, but geometric locality and U(1)U(1)U(1) charge conservation impose an unavoidable transport constraint. We combine exact complementary-channel geometry, charge-sector Haar analysis, and a gate-resolved connected-moment expansion to study one-dimensional covariant encoders under flagged erasure. Charge-Haar codes attain the universal adjacent-charge lower bound up to exponentially small corrections, yielding an exact n−1/2n^{-1/2}n−1/2 extensive-erasure law and a sharp half-erasure transition. For local number-conserving brickwork circuits, diffusion of the logical charge enforces an Ω(n2)Ω(n^2)Ω(n2) encoding-time lower bound; we also prove an O(n3)O(n^3)O(n3) mixing bound for the classical component and reduce the remaining full-channel upper bound to a source-restricted low-support operator-spreading problem. These results identify diffusion as an operational limit on symmetry-constrained quantum coding and establish a route to its exact formation time.