![]() Based on the assumption of scrambling, it was revealed that quantum information is instantly leaked out from the quantum many-body system that models a black hole. The Hayden-Preskill protocol is a qubit-toy model of the black hole information paradox. We illustrate our bounds with two paradigmatic examples, a $N$-qubit mixed GHZ state and a $N$-qubit mixed W state. We model the lack of exact error correction to be equivalent to the action of a single dephasing channel, evaluate the fidelity-based distances both analytically and numerically, and obtain a closed-form expression for any quantum state. With that in mind, we address two distance measures based on the sub- and super-fidelities as a way to bound error approximations, which in turn require lower computational cost. Despite having useful properties, evaluating fidelity measures stand as a challenging task for quantum states with a large number of entangled qubits. In this context, it is common to employ a complementary measure to fidelity as a way to quantify quantum state distinguishability and benchmark approximations in error correction. As a way to circumvent this result, there are several approaches in which one either gives up on exact error correction or continuous symmetries. The Eastin-Knill theorem is a central result of quantum error correction which states that a quantum code cannot correct errors exactly, possess continuous symmetries, and implement a universal set of gates transversely. We also present a type of covariant codes which nearly saturates these lower bounds. Explicit lower bounds are derived for both erasure and depolarizing noises. 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. 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. 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. 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.Ĭovariant 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). We systematically construct codes covariant with respect to general groups, obtaining natural generalizations of qubit codes to, for instance, oscillators and rotors. 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). 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 exhibit codes achieving approximately the same scaling of infidelity with n or d as the lower bound. This bound approaches zero when the number of subsystems n or the dimension d of each subsystem is large. 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. 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. Here, we study the compatibility of these two important principles. 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. ![]()
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