Paper 2023/798

Generalized Hybrid Search and Applications

Abstract

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 recently proposed by Rosmanis. Namely, for an arbitrary distribution $D$ on Boolean functions, the probability that an algorithm equipped with $\tau_c$ classical queries 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 formal security treatment of the Bitcoin protocol called the Bitcoin backbone [Eurocrypt 2015], 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 re-examine the generic security of hash functions [PKC 2016] against quantum-classical 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'' in which the initial state depends on the target distribution $D$. We then show how to analyze its success probability for arbitrary target distributions and, importantly, its optimality for the uniform and the Bernoulli distribution cases.

Note: Updates from previous version of this paper: We have added the design and analysis of fully quantum as well as hybrid algorithms for distributional search, and have shown the optimality for special cases. This supplements the hardness results in the previous version.