Distributed Quantum Error Correction with Permutation-Invariant Approximate Codes

Abstract: Modular quantum computing architectures require error correction schemes that remain effective in the presense of noisy inter-processor operations. We introduce a distributed quantum error correction framework based on approximate codes to address this challenge. Our approach enables concatenation of distinct local codes across modules while allowing logical operations composed primarily of processor-local gates. We derive a lower bound and present corresponding simulations which indicate that this nontraditional approach can provide marked advantage over existing approaches in the highly non-uniform error landscape of a distributed quantum computer. As a concrete realization, we present encoding and decoding circuits for the permutation-invariant W- state code and propose efficient methods for its preparation. These results highlight the potential of approximate distributed error correction strategies for scalable, modular, fault-tolerant quantum computation.