Existence of a polylog-parallel blossom-based MWPM decoder

Determine whether any blossom-algorithm-based minimum-weight perfect matching (MWPM) algorithm can achieve polylogarithmic parallel runtime. Specifically, either construct a blossom-based MWPM decoder with polylogarithmic parallel depth or prove that no blossom-algorithm-based approach can attain polylogarithmic parallel runtime for the detector/path graphs arising in surface-code decoding.

Background

Blossom-based algorithms are the standard practical approach for solving MWPM in surface-code decoding and have seen many optimized implementations. However, their known time complexities remain polynomial, and this scaling can limit decoding throughput in fault-tolerant quantum computation. In contrast, determinant-based approaches have been proposed that achieve polylogarithmic parallel time asymptotically, motivating the question of whether blossom-based methods can match this scaling.

The paper explicitly notes that, despite extensive work on blossom variants and parallelization techniques, there is currently no known blossom-algorithm-based method with polylogarithmic parallel runtime. Establishing whether such an algorithm exists, or proving that it cannot, would clarify the theoretical limits of blossom-based decoding and inform the choice of decoding frameworks for low-overhead FTQC.