Adversarial Quantum Machine Learning: An Information-Theoretic Generalization Analysis

• URL: http://arxiv.org/abs/2402.00176v2
• Date: Thu, 15 Feb 2024 13:18:04 GMT
• Title: Adversarial Quantum Machine Learning: An Information-Theoretic Generalization Analysis
• Authors: Petros Georgiou, Sharu Theresa Jose and Osvaldo Simeone
• Abstract summary: We study the generalization properties of quantum classifiers adversarially trained against bounded-norm white-box attacks. We derive novel information-theoretic upper bounds on the generalization error of adversarially trained quantum classifiers.
• Score: 39.889087719322184
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In a manner analogous to their classical counterparts, quantum classifiers are vulnerable to adversarial attacks that perturb their inputs. A promising countermeasure is to train the quantum classifier by adopting an attack-aware, or adversarial, loss function. This paper studies the generalization properties of quantum classifiers that are adversarially trained against bounded-norm white-box attacks. Specifically, a quantum adversary maximizes the classifier's loss by transforming an input state $\rho(x)$ into a state $\lambda$ that is $\epsilon$-close to the original state $\rho(x)$ in $p$-Schatten distance. Under suitable assumptions on the quantum embedding $\rho(x)$, we derive novel information-theoretic upper bounds on the generalization error of adversarially trained quantum classifiers for $p = 1$ and $p = \infty$. The derived upper bounds consist of two terms: the first is an exponential function of the 2-R\'enyi mutual information between classical data and quantum embedding, while the second term scales linearly with the adversarial perturbation size $\epsilon$. Both terms are shown to decrease as $1/\sqrt{T}$ over the training set size $T$ . An extension is also considered in which the adversary assumed during training has different parameters $p$ and $\epsilon$ as compared to the adversary affecting the test inputs. Finally, we validate our theoretical findings with numerical experiments for a synthetic setting.

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