Quantum Inception Score

TL;DR

The paper introduces the quantum inception score (qIS) as a quantum counterpart to the classical inception score, tying model quality to the Holevo information of the quantum classifier channel. It shows that qIS is bounded below by the cIS and can exceed it when quantum coherence is preserved, inherently linking performance to coherence via the resource theory of asymmetry. The authors demonstrate a quantum advantage from entangled generator outputs and quantify performance degradation due to decoherence through a quantum fluctuation theorem and quantum efficacy. They validate the framework on a quantum phase classification task using a 9-qubit QCNN, illustrating how qIS captures diversity and sharpness in quantum-generated data and how measurement choices affect the classical post-processing impact. Overall, qIS provides a rigorous, information-theoretic metric for assessing quantum generative models and their practical applicability in quantum many-body physics and beyond.

Abstract

Motivated by the great success of classical generative models in machine learning, enthusiastic exploration of their quantum version has recently started. To depart on this journey, it is important to develop a relevant metric to evaluate the quality of quantum generative models; in the classical case, one such example is the (classical) inception score (cIS). In this paper, as a natural extension of cIS, we propose the quantum inception score (qIS) for quantum generators. Importantly, qIS relates the quality to the Holevo information of the quantum channel that classifies a given dataset. In this context, we show several properties of qIS. First, qIS is greater than or equal to the corresponding cIS, which is defined through projection measurements on the system output. Second, the difference between qIS and cIS arises from the presence of quantum coherence, as characterized by the resource theory of asymmetry. Third, when a set of entangled generators is prepared, there exists a classifying process leading to the further enhancement of qIS. Fourth, we harness the quantum fluctuation theorem to characterize the physical limitation of qIS. Finally, we apply qIS to assess the quality of the one-dimensional spin chain model as a quantum generative model, with the quantum convolutional neural network as a quantum classifier, for the phase classification problem in the quantum many-body physics.

Figures (6)

• Figure 1: (a) Four types of protocols in the generative modeling. Generative modeling can involve quantumness in both the generator and classifier (QQ protocol), only in the classifier (CQ protocol), only in the generator (QC protocol), or in neither (CC protocol). This paper studies the QQ protocols. (b) Summary of the main results. We propose a definition of the qIS (Definition \ref{['def:main1']}) and maximally achievable qIS (defined as regularized qIS) by employing the Holevo information and regularized classical capacity, respectively, as a measures of the quality of the quantum generative models. Particularly, when the entangled input $\overline{\rho}_{\rm{in}}$ is allowed, there exists a classifying process $\Phi$ leading to the further quality enhancement. This can be regarded as the quantum advantage in the generative modeling. We also demonstrate that cIS can be recovered via the projection measurements and qIS is larger than or equal to cIS (Theorem \ref{['th:main2']}) due to the presence of quantum coherence (Theorem \ref{['th:main3']}) characterized by the resource theory of asymmetry. Furthermore, by employing the quantum fluctuation theorem approach, we illustrate that the difference between qIS and cIS decreases due to pure dephasing in the quantum classifier, which can be characterized by the quantum efficacy (Theorem \ref{['th:main4']}). (c) An analogous view in the quantum metrology. The ultimate precision limit characterized by the quantum Fisher information (QFI) surpasses the classical limit characterized by the classical Fisher information (CFI). Further enhancement of the QFI can be achieved by the entangled probes, which is the quantum advantage in the metrology.
• Figure 2: The 8-qubit quantum convolutional neural network. $\mathcal{U}_{C}$ and $\mathcal{U}_P$ denote the convolutional and pooling layer. $\{W_1,\cdots, W_7\}$ are the 2-qubit unitaries acting on the pairs of qubits in an alternating manner, which are parameterized by some tunable parameters. $\{K_1,\cdots, K_4\}$ are the 2-qubit unitaries with the form of $K_j\equiv|0\rangle\!\langle 0|\otimes U_j+|1\rangle\!\langle 1|\otimes V_j$, where $U_j$ and $V_j$ are single-qubit unitaries parameterized by some tunable parameters. $\mathcal{T}\equiv {\rm Tr}_{\overline{\mathcal{H}}_S'}$ denotes the partial trace over $\overline{\mathcal{H}}_S'$ (the complement of $\mathcal{H}_S'$). Then, the quantum classifier channel is given by $\Phi\equiv\mathcal{T}\circ\mathcal{U}_P\circ\mathcal{U}_C$.
• Figure 3: The 9-qubit QCNN circuit for the 2-class and 3-class quantum phase classification problems. $T_j$ and $K_j$ denote the 3-qubit gate, and $W_j$ denotes the 2-qubit gate. We take partial trace over all qubits except for the fifth qubit for 2-class and except for both the fifth and eighth qubits for the 3-class classification, respectively.
• Figure 4: The Bloch sphere representation and the histograms of the outputs of the QCNN; panels (a1, a2, a3) show the unbiased case and panels (b1, b2, b3) show the biased case of the 2-class phase classification problem.
• Figure 5: Phase diagram predicted by the QCNN with the projection measurement onto the (a) $X$ axis, (b) $Z$ axis, (c) high-accuracy axis, and (d) optimized axis, for the 2-class classification problem. We use the axes (c) and (d) obtained in the unbiased case.

Theorems & Definitions (7)

• Definition 1: Quantum Inception Score
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof