In this work we first examine the hardness of solving various search problems by hybrid quantum-classical strategies, namely, by algorithms that have both quantum and classical capabilities. We then construct a hybrid quantum-classical search algorithm and analyze its success probability. Regarding the former, for search problems that are allowed to have multiple solutions and in which the input is sampled according to arbitrary distributions we establish their hybrid quantum-classical query complexities -- i.e., given a fixed number of classical and quantum queries, determine what is the probability of solving the search task. At a technical level, our results generalize the framework for hybrid quantum-classical search algorithms proposed by Rosmanis. Namely, for an arbitrary distribution $D$ on Boolean functions, the probability an algorithm equipped with $\tau_c$ classical and $\tau_q$ quantum queries succeeds in finding a preimage of $1$ for a function sampled from $D$ is at most $\nu_D \cdot(2\sqrt{\tau_c} + 2\tau_q + 1)^2$, where $\nu_D$ captures the average (over $D$) fraction of preimages of $1$. As applications of our hardness results, we first revisit and generalize the security of the Bitcoin protocol called the Bitcoin backbone, to a setting where the adversary has both quantum and classical capabilities, presenting a new hybrid honest majority condition necessary for the protocol to properly operate. Secondly, we examine the generic security of hash functions against hybrid adversaries. Regarding our second contribution, we design a hybrid algorithm which first spends all of its classical queries and in the second stage runs a ``modified Grover'' where the initial state depends on the distribution $D$. We show how to analyze its success probability for arbitrary target distributions and, importantly, its optimality for the uniform and the Bernoulli distribution cases.

翻译：本文首先研究了通过量子-经典混合策略（即同时具备量子与经典能力的算法）求解各类搜索问题的困难性，随后构建了一种混合量子-经典搜索算法并分析其成功概率。针对前者，对于允许多个解且输入按任意分布采样的搜索问题，我们确立了其混合量子-经典查询复杂度——即在给定固定数量的经典查询与量子查询条件下，求解搜索任务的概率。在技术层面，我们的成果推广了Rosmanis提出的混合量子-经典搜索算法框架：对任意布尔函数分布$D$，配备$\tau_c$次经典查询和$\tau_q$次量子查询的算法成功找到取自$D$的函数中值为1的原像概率至多为$\nu_D \cdot(2\sqrt{\tau_c} + 2\tau_q + 1)^2$，其中$\nu_D$表示（在$D$上平均的）原像中值为1的比例。作为困难性结论的应用，我们首先重新审视并推广了名为“比特币骨干协议”的比特币协议安全性，将其扩展到对手同时具备量子与经典能力的场景，提出了协议正常运行所需的新型混合诚实多数条件。其次，我们考察了哈希函数对抗混合对手的通用安全性。针对第二项贡献，我们设计了一种混合算法：该算法首先消耗所有经典查询，随后在第二阶段运行“修正版Grover算法”，其初始状态依赖于分布$D$。我们展示了如何分析该算法在任意目标分布下的成功概率，并特别证明了其在均匀分布与伯努利分布情形下的最优性。