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 quantum circuits of shorter depths. 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. Additionally, using this method, we establish a relationship between discrete-time and continuous-time quantum walks, making inroads into a long-standing open problem.

翻译：我们开发了三种实现任意酉算子线性组合（LCU）的新方法。LCU是一种功能强大的量子算法工具，具有多种应用场景。虽然标准LCU过程需要多个辅助量子比特和复杂的多量子比特受控操作，但我们的方法消耗的量子资源显著减少。第一种方法（单辅助量子比特LCU）可在仅需一个辅助量子比特和更浅量子电路深度的情况下，估计由LCU过程制备的任意量子态的可观测量的期望值。第二种方法（模拟LCU）是一种简单、具有物理动机的连续时间模拟LCU方案，专为混合量子-量子模式系统设计。第三种方法（无辅助量子比特LCU）完全不需要辅助量子比特，适用于我们只关心由LCU过程制备的量子态在某个感兴趣子空间中的投影的情况。我们将前两种技术应用于开发一系列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系统。值得注意的是，尽管消耗更少的量子资源，这些算法仍保留了可证明的量子优势。第三种技术使我们能够将离散和连续时间量子游走与其经典对应物联系起来。它还统一了近期在这两种框架下发展的最优量子空间搜索算法，并推动了新算法的开发。此外，利用该方法，我们建立了离散时间与连续时间量子游走之间的关系，为解决一个长期存在的开放问题开辟了新途径。