循环码与解码量子干涉测量
Cycle Codes and Decoded Quantum Interferometry
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中文总结 AI 辅助
本研究分析解码量子干涉测量在循环码上的性能，证明最小权重解码在非二元域上NP困难，并给出经典解码器下的上下界，排除部分量子优势，但发现DQI在正则Max-k-Cut实例上仍能高效实现非平凡满足率。
中文摘要 AI 辅助
解码量子干涉测量（DQI）将具有双变量约束的优化问题简化为循环码的解码问题。对于其中一个此类问题，即最大割问题，先前的工作表明，DQI仅在具有线性围长的图上实现非平凡的满足率保证，而对于这些图，最大割问题在经典计算上是容易的。然而，这些不可能性结果依赖于最小距离假设，这些假设并不代表常见噪声信道的真实可解码性阈值，从而低估了DQI的实际性能。为了估计真实性能，我们在存在不完美解码的情况下，针对固定实例推导了DQI的满足率保证，并推广了针对随机实例的先前结果。随后，我们研究了任意有限域上的同调循环码，并证明了对于每个大小为$q>2$的域，最小权重解码是NP困难的，这与二元循环码的已知结果形成对比。在具有对数围长的Linial--Simkin正则图系综中，我们证明了最大似然恢复界以及一族非均匀加性噪声信道的强逆定理。此外，LP解码器能够以多项式时间恢复随机定位错误的正比例部分，且错误值任意。这些结果使我们能够证明，当DQI限制使用经典解码器时，其可达到的近似最优值的上下界。这些界排除了我们所分析机制中的量子优势，但也识别出一族正则Max-$k$-Cut实例，在这些实例上，DQI高效地实现了非平凡的满足率保证。
英文摘要
Decoded Quantum Interferometry (DQI) reduces optimization problems with two-variable constraints to decoding cycle codes. For one such problem, namely MaxCut, prior work showed that DQI achieves a nontrivial satisfaction fraction guarantee only on linear-girth graphs, for which MaxCut is classically easy. However, these no-go results rely on minimum distance assumptions that do not represent true decodability thresholds for common noise channels, thus underestimating actual DQI performance. To estimate the true performance, we derive DQI satisfaction guarantees in the presence of imperfect decoding for fixed instances and generalize prior results for random instances. We then study homological cycle codes over arbitrary finite fields, and show that minimum-weight decoding is NP-hard for every field of size $q>2$, in contrast with known results for binary cycle codes. On the Linial--Simkin ensemble of regular graphs with logarithmic girth, we prove maximum-likelihood recovery bounds and a strong converse for a family of non-uniform additive noise channels. Moreover, LP decoders provide polynomial-time recovery of a positive fraction of randomly located errors with arbitrary values. These results allow us to prove upper and lower bounds on the approximate optima achievable by DQI when restricted to classical decoders. These bounds rule out quantum advantage in the regimes we analyze but also identify a family of regular Max-$k$-Cut instances on which DQI efficiently achieves a nontrivial satisfaction fraction guarantee.
发表机构
- Global Technology Applied Research, JPMorganChase（摩根大通全球技术应用研究）
- Harvard University（哈佛大学）
- Google Quantum AI（谷歌量子人工智能）
- Sandia National Laboratories（桑迪亚国家实验室）
机构由 AI 辅助整理，请以论文原文为准。
