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Estimating the difference between quantum data is crucial in quantumcomputing. However, as typical characterizations of quantum data similarity,the trace distance and quantum fidelity are believed to be exponentially-hardto evaluate in general. In this work, we introduce hybrid quantum-classicalalgorithms for these two distance measures on near-term quantum devices whereno assumption of input state is required. First, we introduce the VariationalTrace Distance Estimation (VTDE) algorithm. We in particular provide thetechnique to extract the desired spectrum information of any Hermitian matrixby local measurement. A novel variational algorithm for trace distanceestimation is then derived from this technique, with the assistance of a singleancillary qubit. Notably, VTDE could avoid the barren plateau issue withlogarithmic depth circuits due to a local cost function. Second, we introducethe Variational Fidelity Estimation (VFE) algorithm. We combine Uhlmann'stheorem and the freedom in purification to translate the estimation task intoan optimization problem over a unitary on an ancillary system with fixedpurified inputs. We then provide a purification subroutine to complete thetranslation. Both algorithms are verified by numerical simulations andexperimental implementations, exhibiting high accuracy for randomly generatedmixed states.

Variational Quantum Algorithms for Trace Distance and Fidelity Estimation