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：我们发展了三种新方法来实现任意酉线性组合（LCU），这是一种强大的量子算法工具，具有广泛的应用。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，但我们提出的方法消耗的量子资源显著更少。第一种方法（单辅助量子比特LCU）能够估计由LCU过程制备的任何量子态下可观测量的期望值，同时仅需一个辅助量子比特，且无需多量子比特受控操作。第二种方法（模拟LCU）是一种简单、物理驱动的连续时间模拟LCU，专为混合量子比特-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，在仅关注LCU过程制备的量子态在某子空间中的投影时尤为有用。我们将前两种技术应用于一系列实际问题，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，从而开发出新的量子算法。值得注意的是，尽管消耗更少的量子资源，这些算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。它还统一了这两种框架中最近发展的最优量子空间搜索算法，并推动了需要更少辅助量子比特的新算法的开发。总体而言，我们的结果具有高度普适性，可轻易应用于其他问题，甚至超出本文所考虑的范围。