Fault-tolerant Preparation of Distant Logical Bell Pair -- with application in the magic square game

Measures of quantum nonlocality traditionally assume perfect local computation. In real experiments, however, each computational primitive is imperfect. Fault-tolerant techniques enable arbitrarily accurate quantum computation but do not necessarily preserve optimized measures of nonlocality. We examine the impact of low noise on quantum nonlocality in nonlocal games, where even small imperfections can disproportionately increase entanglement consumption. Focusing on the fault-tolerant magic square game, we optimize the tradeoff between noisy entanglement consumption and deficit in the game value. We introduce an interface circuit and logical entanglement purification protocol (EPP) to efficiently translate states between physical and logical qubits and purify noisy logical Bell pair, reducing Bell pair consumption. Our analytical and numerical results, particularly for the $[[7^k,1,3^k]]$ concatenated Steane code, demonstrate exponential Bell pair savings and a higher noise threshold. We establish theoretical lower bounds for local noise threshold of $4.70\times10^{-4}$ and an initial Bell pair infidelity threshold of $18.3\%$. Our framework is adaptable to various quantum error-correcting codes (QECCs) and experimental platforms. This work not only advances fault-tolerant nonlocal games but also inspires further research on interfacing different QECCs, fostering modular quantum architectures and the quantum internet.