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要通过辨别性措施来判断量子信息处理协议的性能,这种措施量化了协议的实际结果如何区别于理想情况。最突出的辨别性措施是那些基于真实和痕量距离的辨别性与物理解释。在本文件中,我们提出并审查了几种根据痕量距离和真实性估计可辨别性尺度的算法。这些算法可用于区分量子状态、渠道和战略(文献中最后一种也称为“量子梳”)。基于忠诚的算法对这些可辨别性措施提供了新的物理解释,即单个验证人(或相互竞争的证明人)能够说服验证人接受相关计算结果的最大概率。我们用参数化量子电路的变法模拟了许多这些算法。我们发现,模拟在无噪音和噪音的情景中都非常接近所有考虑的例子。此外,噪音模拟显示了参数的噪声弹性。最后,我们在各种量计算复杂类别和距离估计问题之间建立了强有力的关系。