Transition amplitudes and transition probabilities are relevant to many areas of physics simulation, including the calculation of response properties and correlation functions. These quantities can also be related to solving linear systems of equations. Here we present three related algorithms for calculating transition probabilities. First, we extend a previously published short-depth algorithm, allowing for the two input states to be non-orthogonal. Building on this first procedure, we then derive a higher-depth algorithm based on Trotterization and Richardson extrapolation that requires fewer circuit evaluations. Third, we introduce a tunable algorithm that allows for trading off circuit depth and measurement complexity, yielding an algorithm that can be tailored to specific hardware characteristics. Finally, we implement proof-of-principle numerics for models in physics and chemistry and for a subroutine in variational quantum linear solving (VQLS). The primary benefits of our approaches are that (a) arbitrary non-orthogonal states may now be used with small increases in quantum resources, (b) we (like another recently proposed method) entirely avoid subroutines such as the Hadamard test that may require three-qubit gates to be decomposed, and (c) in some cases fewer quantum circuit evaluations are required as compared to the previous state-of-the-art in NISQ algorithms for transition probabilities.

翻译：跃迁振幅与跃迁概率在诸多物理模拟领域（包括响应性质与关联函数的计算）中具有重要意义。这些物理量还能与线性方程组的求解建立联系。本文提出三种相关的跃迁概率计算算法。首先，我们扩展了此前发表的浅层电路算法，使其能够处理非正交的两种输入态。基于该第一类过程，我们随后推导出一种基于特罗特分解与理查森外推的高深度算法，该算法所需电路评估次数更少。第三，我们引入一种可调谐算法，允许在电路深度与测量复杂度之间进行权衡，从而得到一种能针对特定硬件特性进行定制的算法。最后，我们针对物理学与化学模型，以及变分量子线性求解（VQLS）中的子程序，完成了原理验证数值实验。我们方法的主要优势在于：(a) 现在可以使用任意非正交态，且只需少量增加量子资源；(b) 我们（如同另一近期提出的方法）完全避免了哈达玛测试等可能需要分解三量子比特门的子程序；(c) 在某些情况下，与先前最先进的NISQ用于计算跃迁概率的算法相比，所需量子电路评估次数更少。