Quantum relative entropy serves as a fundamental measure of the distinguishability between quantum states and plays a pivotal role in quantum information science. Despite its importance, efficiently estimating quantum relative entropy between two quantum states on quantum computers remains a significant challenge. In this work, we propose a quantum algorithm for directly estimating quantum relative entropy and Petz Rényi divergence from two unknown quantum states on quantum computers. The circuit size of our algorithm is at most 2n + 1 with n being the number of qubits in the quantum states and it is directly applicable to distributed scenarios, where quantum states to be compared are hosted on cross-platform devices. We validate the effectiveness of our method through numerical experiments and observe the absence of the barren plateau phenomenon. As an application, we employ our algorithm to investigate the superadditivity of quantum channel capacity. Numerical simulations reveal new examples of qubit channels exhibiting strict superadditivity of coherent information, highlighting the potential of quantum machine learning to address quantum-native problems. Quantum relative entropy is a central measure of distinguishability between quantum states, yet its estimation on quantum computers remains challenging. Here, the authors propose a variational quantum algorithm that directly evaluates it from unknown states and works across distributed quantum devices.

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A new variational quantum algorithm is proposed for directly estimating quantum relative entropy and Petz Rényi divergence from two unknown quantum states on quantum computers. The algorithm uses circuits of at most 2n+1 qubits, avoids the barren plateau problem, and supports distributed quantum devices across platforms. It is applied to study superadditivity of quantum channel capacity, uncovering new examples of qubit channels with strict superadditivity of coherent information.

Estimating quantum relative entropies on quantum computers