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

We present a simple proof of the approximate Eastin-Knill theorem, which connects the quality of a quantum error-correcting code (QECC) with its ability to achieve a universal set of transversal logical gates. Our derivation employs powerful bounds on the quantum Fisher information in generic quantum metrological protocols to characterize the QECC performance measured in terms of the worst-case entanglement fidelity. The theorem is applicable to a large class of decoherence models, including erasure and depolarizing noise. Our approach is unorthodox, as instead of following the established path of utilizing QECCs to mitigate noise in quantum metrological protocols, we apply methods of quantum metrology to explore the limitations of QECCs.

Figure 1

(a) A generic quantum metrological protocol involving $m$ sequential applications of the parameter encoding channel ${⨂}_{i=1}^n({\mathcal{N}}_{A_i}\circ {\mathcal{U}}_{A_i}^{\theta })$ interleaved with the control operations. (b) A metrological protocol with the initial state ${\rho }_A={\mathcal{E}}_{A\to L}({\rho }_L)$ encoded into a QECC and the control corresponding to a composition of the recovery ${\mathcal{R}}_{A\to L}$ and encoding ${\mathcal{E}}_{A\to L}$ maps. (c) Using the covariance property of ${\mathcal{E}}_{L\to A}$, we find an equivalent scheme where the parameter-dependent unitary ${⨂}_{i=1}^n{U}_{A_i}^{\theta }$ on the physical system $A$ (shaded in gray) is expressed as $U_L^{\theta }$ on the logical system $L$ (shaded in green).