The performance of a quantum information processing protocol is ultimately judged by distinguishability measures that quantify how distinguishable the actual result of the protocol is from the ideal case. The most prominent distinguishability measures are those based on the fidelity and trace distance, due to their physical interpretations. In this paper, we propose and review several algorithms for estimating distinguishability measures based on trace distance and fidelity. The algorithms can be used for distinguishing quantum states, channels, and strategies (the last also known in the literature as ``quantum combs''). The fidelity-based algorithms offer novel physical interpretations of these distinguishability measures in terms of the maximum probability with which a single prover (or competing provers) can convince a verifier to accept the outcome of an associated computation. We simulate many of these algorithms by using a variational approach with parameterized quantum circuits. We find that the simulations converge well in both the noiseless and noisy scenarios, for all examples considered. Furthermore, the noisy simulations exhibit a parameter noise resilience. Finally, we establish a strong relationship between various quantum computational complexity classes and distance estimation problems.

翻译：量子信息处理协议的性能最终由区分度度量来判断，这些度量量化了协议的实际结果与理想情况的可区分程度。最突出的区分度度量是基于保真度和迹距离的度量，这源于其物理诠释。在本文中，我们提出并回顾了几种基于迹距离和保真度估计区分度度量的算法。这些算法可用于区分量子态、通道和策略（最后一种在文献中也称为“量子梳”）。基于保真度的算法为这些区分度度量提供了新的物理诠释，即单个证明者（或竞争证明者）能够说服验证者接受相关计算结果的极大概率。我们通过使用参数化量子电路的变分方法模拟了其中许多算法。我们发现，对于所有考虑的示例，模拟在无噪和有噪场景下均能良好收敛。此外，有噪模拟展现出参数噪声鲁棒性。最后，我们建立了各种量子计算复杂度类与距离估计问题之间的强关联。