Estimating the difference between quantum data is crucial in quantum computing. In particular, trace distance and quantum fidelity are vital for verifying various quantum information processing tasks. In this work, we introduce hybrid quantum-classical algorithms for practical distance measure estimation on near-term quantum devices. First, we introduce the Variational Trace Distance Estimation (VTDE) algorithm. We in particular show that local measurement results can extract the desired spectrum information of any Hermitian matrix. We further exploit this fact to design a novel variational algorithm for trace norm estimation that only involves one ancillary qubit. Notably, the cost function in VTDE gathers information from a single-qubit observable and thus could avoid the barren plateau issue with logarithmic depth parameterized circuits. Second, we introduce the Variational Fidelity Estimation (VFE) algorithm. We combine Uhlmann's theorem and the freedom in purification to translate the estimation task into an optimization problem over a unitary on an ancillary system with fixed purified inputs. We then introduce a purification subroutine to complete the task. Numerical experiments for both algorithms have been conducted to show the validity of our methods. The metrics are estimated with high accuracy for randomly generated mixed states.

翻译：估计量子数据之间的差异在量子计算中至关重要。 特别是, 追踪距离和量的可靠性对于核实各种量子信息处理任务至关重要。 在这项工作中, 我们采用混合量子古典算法对短期量子装置进行实际的距离估测。 首先, 我们引入变异追踪距离估计算法。 我们特别显示, 本地测量结果可以提取任何Hermitian 矩阵中想要的频谱信息。 我们进一步利用这一事实来设计一个新的变异算法, 用于追踪标准估计, 仅涉及一个辅助方位。 值得注意的是, VTDE 的成本函数从一个可观测的单方位中收集信息, 从而可以避免用对数深度参数化的电路路段进行贫瘠高地高地层问题。 其次, 我们引入变异偏差测算算法。 我们结合了乌尔曼的理论和净化自由, 将估算任务转化为一个仅包含固定净化投入的辅助系统的统一的优化问题。 然后我们引入一个净化子系统的成本函数来完成这项任务。 用于测量我们两种测算的精度的精度的精度的精度的精度, 随机测方法的精确度是随机的。 。