Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro's Search with Advice [Montanaro, TQC 2010].

翻译：用于函数评估的量子跨度程序算法有时在承诺输入具有特定结构时能够降低查询复杂度。我们设计了一种改进的跨度程序算法，表明即使在没有预先承诺的情况下，这些改进依然存在，并将此方法扩展到更一般的状态转换问题。作为应用，我们证明了在某些搜索问题中平均查询复杂度上的指数级和超多项式级量子优势，推广了Montanaro的“带建议的搜索”[Montanaro, TQC 2010]。