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：跃迁振幅和跃迁概率与物理模拟的众多领域密切相关，包括响应性质和关联函数的计算。这些量还可与线性方程组的求解相关联。本文提出三种用于计算跃迁概率的关联算法。首先，我们扩展了先前发表的短深度算法，使其容许两个输入态非正交。基于这一初步流程，我们推导出基于特罗特分解和理查森外推的高深度算法，该算法需要更少的电路评估。第三，我们引入了一种可调谐算法，允许在电路深度与测量复杂度之间进行权衡，从而得到可针对特定硬件特性定制的算法。最后，我们对物理和化学模型以及变分量子线性求解（VQLS）的子系统进行了原理验证数值模拟。我们方法的主要优势在于：(a) 任意非正交态现在仅需少量增加量子资源即可使用；(b) （与另一种近期提出的方法类似）我们完全避免了可能需分解为三量子比特门的子例程（如Hadamard测试）；(c) 在某些情况下，相比先前用于跃迁概率的NISQ算法的现有技术，所需的量子电路评估次数更少。