Estimating quantum relative entropies on quantum computers

TL;DR

The paper tackles the challenge of estimating quantum relative entropy and Petz Renyi divergences between unknown quantum states on quantum hardware. It introduces a variational framework that expresses these quantities as quadrature-averaged standard $f$-divergences, paired with a novel Hermitian-polynomial parameterization to enable efficient trace estimation via quantum circuits. The approach yields operator-convex loss landscapes, trainable circuits, and rigorous error bounds, with demonstrated accuracy in numerical experiments and a compelling application to identifying superadditivity of quantum channel capacity. The framework supports distributed, cross-platform quantum comparisons and paves the way for quantum-native information-theoretic tasks on near-term devices.

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.

Figures (15)

• Figure 1: Estimating quantum relative entropy using a variational quantum algorithm. Identical copies of the quantum states, possibly hosted on different quantum computers, are processed through parameterized quantum circuits to extract relevant information. The variational algorithm estimates quantum $f$-divergences, and the quantum relative entropy is computed as a weighted average of these $f$-divergences, determined by the quadrature nodes and weights. The estimation of Petz Rényi divergence follows a similar procedure.
• Figure 2: Performance of GRJ quadrature approximation. (a) Quadrature approximation for the function $y = \log x$; (b) Quadrature approximation for the function $y = x^{-0.5}$. The approximation converges to the exact value as the number of quadrature nodes $m$ increases.
• Figure 3: Visualization of unitary operation $\chi_{\bm{i}}$. We evolve the input state $|0\rangle\langle0|\otimes \rho\otimes |\bm{j}\rangle\langle\bm{j}|$ with a unitary operation $\chi_{\bm{i}}$, and then measure the first qubit in the computational basis. The probability of outcome $0$ of the measurement gives the value of $(\Theta_{\bm{j},\bm{i}}+1)/2$.
• Figure 4: Visualization of unitary operation $\chi$. It consists of two Hadamard gates $H$, controlled $U_{\bm{\theta}}$, $V_{\bm{\beta}}$ gates, and a controlled swap gates.
• Figure 5: Ansatz for testing barren plateau phenomenon.

Theorems & Definitions (15)

• Definition 1: Standard quantum $f$-divergence
• Lemma 2: brown2024device
• Lemma 3
• Lemma 4
• proof
• Proposition 5: Error bound for estimating $D(\rho\|\sigma)$
• proof
• Lemma 6: faust2023rational
• Lemma 7: Error bound for estimating $Q_\alpha(\rho\|\sigma)$
• proof