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 ($\textit{Single-Ancilla LCU}$) 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 ($\textit{Analog LCU}$) is a simple, physically motivated, continuous-time analogue of LCU, tailored to hybrid qubit-qumode systems. The third method ($\textit{Ancilla-free LCU}$) 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.

中文翻译：

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在中期量子计算机上实现 Unitaries 的任意线性组合


我们开发了三种新方法来实现任何线性幺正组合 （LCU），这是一种具有多种应用的强大量子算法工具。虽然标准 LCU 程序需要多个辅助量子比特和复杂的多量子比特控制操作，但我们的方法消耗的量子资源要少得多。第一种方法 （$\textit{Single-Ancilla LCU}$） 估计可观察对象相对于 LCU 过程准备的任何量子状态的期望值，同时只需要一个辅助量子比特，不需要多量子比特控制操作。第二种方法 （$\textit{Analog LCU}$） 是一种简单的、物理驱动的 LCU 连续时间模拟，专为混合量子比特-量子模式系统量身定制。第三种方法 （$\textit{Ancilla-free LCU}$） 根本不需要辅助量子比特，当我们对量子态（由 LCU 过程准备）在某个感兴趣的子空间中的投影感兴趣时非常有用。我们应用前两种技术为广泛的实际问题开发新的量子算法，包括哈密顿模拟、基态准备和属性估计以及量子线性系统。值得注意的是，尽管消耗的量子资源较少，但它们保留了可证明的量子优势。第三种技术允许我们将离散和连续时间量子游走与它们的经典对应物联系起来。它还在这两个框架中统一了最近开发的最佳量子空间搜索算法，并导致了需要更少辅助量子比特的新算法的开发。总的来说，我们的结果非常通用，可以很容易地应用于其他问题，甚至超出这里考虑的问题。