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We present a simple proof of the approximate Eastin-Knill theorem, whichconnects the quality of a quantum error-correcting code (QECC) with its abilityto achieve a universal set of transversal logical gates. Our derivation employspowerful bounds on the quantum Fisher information in generic quantummetrological protocols to characterize the QECC performance measured in termsof the worst-case entanglement fidelity. The theorem is applicable to a largeclass of decoherence models, including independent erasure and depolarizingnoise. Our approach is unorthodox, as instead of following the established pathof utilizing QECCs to mitigate noise in quantum metrological protocols, weapply methods of quantum metrology to explore the limitations of QECCs.

Using Quantum Metrological Bounds in Quantum Error Correction: A Simple Proof of the Approximate Eastin-Knill Theorem