Over the years, the framework of Linear combination of unitaries (LCU) has been extremely useful for designing a plethora of quantum algorithms. In this work, we explore whether this widely applicable paradigm can be implemented on quantum computers that will be available immediately after the current NISQ stage. To this end, we develop three variants of LCU and apply each, to quantum algorithms of practical interest. First, we develop a physically motivated, continuous-time analogue of LCU (``Analog LCU''). This technique, implementable on hybrid qubit-qumode systems, is simpler than its discrete-time counterpart. We use this method to develop analog quantum algorithms for ground state preparation and quantum linear systems. We also develop a randomized quantum algorithm to sample from functions of Hamiltonians applied to quantum states (``Single-Ancilla LCU''). This approach repeatedly samples from a short-depth quantum circuit and uses only a single ancilla qubit. We use this to estimate expectation values of observables in the ground states of a Hamiltonian, and in the solution of quantum linear systems. This method is suitable for early fault-tolerant quantum computers. Our third approach stems from the observation that for several applications, it suffices to replace LCU with randomized sampling of unitaries according to the distribution of the LCU coefficients (``Ancilla-free LCU''). This is particularly useful when one is interested in the projection of a quantum state implemented by an LCU procedure in some subspace of interest. We demonstrate that this technique applies to the spatial search problem and helps establish a relationship between discrete and continuous-time quantum walks with their classical counterparts. Our work demonstrates that generic quantum algorithmic paradigms, such as LCU, can potentially be implemented on intermediate-term quantum devices.

翻译：多年来，单位矩阵线性组合（LCU）框架在大量量子算法的设计中展现出极高的实用性。本研究旨在探索这一广泛适用的范式能否在超越当前NISQ时代后立即可用的量子计算机上实现。为此，我们开发了三种LCU变体，并将其分别应用于具有实际意义的量子算法。首先，我们提出了一种基于物理直觉的LCU连续时间模拟方案（"模拟LCU"）。该技术可在混合量子-量子模系统上实现，比其离散时间版本更为简洁。我们利用该方法开发了用于基态制备和量子线性系统的模拟量子算法。其次，我们设计了一种随机化量子算法，用于从作用于量子态的哈密顿量函数中采样（"单辅助量子比特LCU"）。该方法通过重复采样短深度量子电路，仅需单个辅助量子比特即可实现。我们以此估计哈密顿量基态可观测量期望值及量子线性系统解，适用于早期容错量子计算机。第三种方法源于一个观察：对于某些应用场景，只需根据LCU系数的分布对单位矩阵进行随机采样（"无辅助量子比特LCU"）即可替代完整LCU。该方法特别适用于需要将LCU过程实现的量子态投影到特定子空间的情形。我们证明了该技术可应用于空间搜索问题，并建立了离散与连续时间量子行走与其经典对应物之间的联系。本研究表明，LCU等通用量子算法范式有望在中期量子设备上实现。