Estimating the difference between quantum data is crucial in quantum computing. However, as typical characterizations of quantum data similarity, the trace distance and quantum fidelity are believed to be exponentially-hard to evaluate in general. In this work, we introduce hybrid quantum–classical algorithms for these two distance measures on near-term quantum devices where no assumption of input state is required. First, we introduce the variational trace distance estimation (VTDE) algorithm. We in particular provide the technique to extract the desired spectrum information of any Hermitian matrix by local measurement. A novel variational algorithm for trace distance estimation is then derived from this technique, with the assistance of a single ancillary qubit. Notably, VTDE could avoid the barren plateau issue with logarithmic depth circuits due to a local cost function. Second, we introduce the variational fidelity estimation 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 provide a purification subroutine to complete the translation. Both algorithms are verified by numerical simulations and experimental implementations, exhibiting high accuracy for randomly generated mixed states.

估计量子数据之间的差异在量子计算中至关重要。然而，作为量子数据相似性的典型表征，通常认为轨迹距离和量子保真度是指数级难以评估的。在这项工作中，我们在不需要假设输入状态的近期量子设备上为这两个距离测量引入了混合量子经典算法。首先，我们介绍变分迹线距离估计（VTDE）算法。我们特别提供了通过局部测量提取任何 Hermitian 矩阵的所需光谱信息的技术。然后，在单个辅助量子比特的帮助下，从该技术推导出了一种用于轨迹距离估计的新变分算法。尤其，由于局部成本函数，VTDE 可以避免对数深度电路的贫瘠高原问题。其次，我们介绍了变分保真度估计算法。我们将 Uhlmann 定理和纯化自由度结合起来，将估计任务转化为对具有固定纯化输入的辅助系统上的酉优化问题。然后我们提供一个纯化子程序来完成翻译。两种算法都通过数值模拟和实验实现验证，对随机生成的混合状态表现出高精度。我们将 Uhlmann 定理和纯化自由度结合起来，将估计任务转化为对具有固定纯化输入的辅助系统上的酉优化问题。然后我们提供一个纯化子程序来完成翻译。两种算法都通过数值模拟和实验实现验证，对随机生成的混合状态表现出高精度。我们将 Uhlmann 定理和纯化自由度结合起来，将估计任务转化为对具有固定纯化输入的辅助系统上的酉优化问题。然后我们提供一个纯化子程序来完成翻译。两种算法都通过数值模拟和实验实现验证，对随机生成的混合状态表现出高精度。