We develop three new methods to implement any Linear Combination of Unitaries (LCU), a powerful quantum algorithmic tool with diverse applications. While the standard LCU procedure requires several ancilla qubits and sophisticated multi-qubit controlled operations, our methods consume significantly fewer quantum resources. The first method (Single-Ancilla LCU) estimates expectation values of observables with respect to any quantum state prepared by an LCU procedure while requiring only a single ancilla qubit, and no multi-qubit controlled operations. The second approach (Analog LCU) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method (Ancilla-free LCU) requires no ancilla qubit at all and is useful when we are interested in the projection of a quantum state (prepared by the LCU procedure) in some subspace of interest. We apply the first two techniques to develop new quantum algorithms for a wide range of practical problems, ranging from Hamiltonian simulation, ground state preparation and property estimation, and quantum linear systems. Remarkably, despite consuming fewer quantum resources they retain a provable quantum advantage. The third technique allows us to connect discrete and continuous-time quantum walks with their classical counterparts. It also unifies the recently developed optimal quantum spatial search algorithms in both these frameworks, and leads to the development of new ones that require fewer ancilla qubits. Overall, our results are quite generic and can be readily applied to other problems, even beyond those considered here.

翻译：我们提出了三种新方法来实现任意酉算子线性组合（LCU）——一种具有多样化应用的强大量子算法工具。标准LCU流程需要多个辅助量子比特和复杂的多量子比特受控操作，而我们的方法显著减少了所需的量子资源。第一种方法（单辅助比特LCU）仅需单个辅助量子比特且无需多量子比特受控操作，即可估计通过LCU流程制备的任意量子态相对于可观测量期望值。第二种方法（模拟LCU）是LCU的简单、物理启发的连续时间模拟版本，专为混合量子比特-量子模系统设计。第三种方法（无辅助比特LCU）完全不需要辅助量子比特，适用于我们仅关注通过LCU流程制备的量子态在特定子空间投影的情形。我们将前两种技术应用于开发针对广泛实际问题的量子算法，包括哈密顿量模拟、基态制备与性质估计以及量子线性方程组求解。值得注意的是，尽管消耗更少量子资源，这些方法仍保持可证明的量子优势。第三种技术使我们能够将离散与连续时间量子行走与其经典对应建立联系。该方法不仅统一了近期在这两种框架下开发的最优量子空间搜索算法，还催生了需要更少辅助量子比特的新算法。总体而言，我们的结果具有高度通用性，可便捷应用于其他问题，甚至超越本文讨论的范畴。