Estimating quantum relative entropies on quantum computers

TL;DR

This paper introduces the first quantum algorithm for directly estimating quantum relative entropy and Petz Renyi divergence between unknown quantum states, with applications in quantum information and machine learning.

Contribution

The authors develop a novel quantum algorithm with a circuit size of at most 2n+1 qubits, capable of estimating quantum relative entropy directly on quantum computers, including distributed scenarios.

Findings

Algorithm successfully estimates quantum relative entropy and Petz Renyi divergence.

Numerical experiments show the absence of barren plateau phenomenon.

Application to quantum channel capacity reveals new superadditivity examples.

Abstract

Quantum relative entropy, a quantum generalization of the renowned Kullback-Leibler divergence, 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 the first quantum algorithm for directly estimating quantum relative entropy and Petz Renyi divergence from two unknown quantum states on quantum computers, addressing open problems highlighted in [Phys. Rev. A 109, 032431 (2024)] and [IEEE Trans. Inf. Theory 70, 5653-5680 (2024)]. Notably, 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 quantum computers. We prove that our loss function is operator-convex, ensuring that any local minimum is also a global minimum. 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.