Security in machine learning is fragile when data are exfiltrated or perturbed, yet existing frameworks rarely connect the definition and analysis of the security to learnability. In this work, we develop a theory of secure learning grounded in the probably-approximately-correct (PAC) viewpoint and develop an operational framework that links data-path behavior to finite-sample budgets. In our formulation, an accuracy-confidence target is evaluated via a run-based sequential test that halts after a prescribed number of consecutive validations, and a closed-form budget bound guarantees the learning success if the data-path channel is admissible; the acceptance must also exceed a primitive random-search baseline. We elevate and complete our secure-learning construction in the context of quantum information -- establishing quantum-secure PAC learning: for prepare-and-measure scenarios, the data-path admissibility is set to be threshold fixed by Holevo information, not a learner-tunable tolerance. Thus, a certified information advantage for the learner directly becomes the learning security -- an effect with no classical analogue. The channel-determined confidence follows naturally and basis sifting is incorporated for practical deployments. This is the first complete framework that simultaneously embeds a security notion and an operational sample-budget law within the PAC learning and anchors the security in quantum information. The resulting blueprint points toward standardized guarantees for the learning security, with clear avenues for PAC-Bayes extensions and for integration with advanced quantum machine learning front ends.
Security in machine learning is fragile when data are exfiltrated or perturbed, yet existing frameworks rarely connect the definition and analysis of the security to learnability. In this work, we develop a theory of secure learning grounded in the probably-approximately-correct (PAC) viewpoint and develop an operational framework that links data-path behavior to finite-sample budgets. In our formulation, an accuracy-confidence target is evaluated via a run-based sequential test that halts after a prescribed number of consecutive validations, and a closed-form budget bound guarantees the learning success if the data-path channel is admissible; the acceptance must also exceed a primitive random-search baseline. We elevate and complete our secure-learning construction in the context of quantum information -- establishing quantum-secure PAC learning: for prepare-and-measure scenarios, the data-path admissibility is set to be threshold fixed by Holevo information, not a learner-tunable tolerance. Thus, a certified information advantage for the learner directly becomes the learning security -- an effect with no classical analogue. The channel-determined confidence follows naturally and basis sifting is incorporated for practical deployments. This is the first complete framework that simultaneously embeds a security notion and an operational sample-budget law within the PAC learning and anchors the security in quantum information. The resulting blueprint points toward standardized guarantees for the learning security, with clear avenues for PAC-Bayes extensions and for integration with advanced quantum machine learning front ends.
Studies addressing the question Can a learner complete the learning securely? have recently been spurred from the standpoints of fundamental theory and potential applications. In the relevant context of this question, we present a classical-quantum hybrid sampling protocol and define a security condition that allows only legitimate learners to prepare a finite set of samples that guarantees the success of the learning; the security condition excludes intruders. We do this by combining our security concept with the bound of the so-called probably approximately correct (PAC) learning, and therefore, we call the condition quantum secure PAC learning condition. We show that while the lower bound on the learning samples guarantees PAC learning, an upper bound can be derived to rule out adversarial learners. Such a secure learning condition is appealing, because it is defined only by the size of samples required for the successful learning and is independent of the algorithm employed. Notably, the security stems from the fundamental quantum no-broadcasting principle. No such condition can thus occur in any classical regime, where learning samples can be copied. Owing to the hybrid architecture, our scheme also offers a practical advantage for implementation in noisy intermediate-scale quantum (NISQ) devices.
Whether quantum computation can reduce the amount of data required for conventional machine learning is a central problem in quantum machine learning. In probably approximately correct (PAC) learning, a learner receives training examples sampled from an unknown probability distribution. Quantum PAC learning extends this framework by allowing access to quantum examples, which are quantum states that coherently encode this distribution, or to unitaries that prepare quantum examples, with or without inverse access. When only copies of quantum examples are available, the optimal worst-case sample complexity asymptotically matches that of classical learning. In contrast, access to both a preparation unitary and its inverse can improve the accuracy dependence of query complexity. However, it has remained open whether access to the preparation unitary alone suffices for this improvement.
In this work, we resolve this question by determining the optimal query complexity with forward-only access, taking the worst case over compatible preparation unitaries and their dimensions. In both realizable and agnostic PAC learning, this complexity matches, up to constant factors, the optimal sample complexity achievable with classical examples or copies of quantum examples. To prove these results, we construct, for any algorithm with forward-only access, a simulator that approximates the algorithm's Haar-averaged output state using as many copies of the prepared state as the algorithm makes queries. These results show that inverse access is essential for the known improvement in accuracy dependence in this worst-case setting. Beyond PAC learning, our reduction provides a general tool for transferring lower bounds for state copies to query lower bounds for forward-only access to the corresponding preparation unitaries.
Studies addressing the question ``Can a learner complete the learning securely?'' have recently been spurred from the standpoints of fundamental theory and potential applications. In the relevant context of this question, we present a classical-quantum hybrid sampling protocol and define a security condition that allows only legitimate learners to prepare a finite set of samples that guarantees the success of the learning; the security condition excludes intruders. We do this by combining our security concept with the bound of the so-called probably approximately correct (PAC) learning. We show that while the lower bound on the learning samples guarantees PAC learning, an upper bound can be derived to rule out adversarial learners. Such a secure learning condition is appealing, because it is defined only by the size of samples required for the successful learning and is independent of the algorithm employed. Notably, the security stems from the fundamental quantum no-broadcasting principle. No such condition can thus occur in any classical regime, where learning samples can be copied. Owing to the hybrid architecture, our scheme also offers a practical advantage for implementation in noisy intermediate-scale quantum devices.
Understanding the power of quantum data in machine learning is central to many proposed applications of quantum technologies. While access to quantum data can offer exponential advantages for carefully designed learning tasks and often under strong assumptions on the data distribution, it remains an open question whether such advantages persist in less structured settings and under more realistic, naturally occurring distributions. Motivated by these practical concerns, we introduce a systematic framework based on quantum lossy data compression to bound the power of quantum data in the context of probably approximately correct (PAC) learning. Specifically, we provide lower bounds on the sample complexity of quantum learners for arbitrary functions when data is drawn from Zipf's distribution, a widely used model for the empirical distributions of real-world data. We also establish lower bounds on the size of quantum input data required to learn linear functions, thereby proving the optimality of previous positive results. Beyond learning theory, we show that our framework has applications in secure delegated quantum computation within the measurement-based quantum computation (MBQC) model. In particular, we constrain the amount of private information the server can infer, strengthening the security guarantees of the delegation protocol proposed in (Mantri et al., PRX, 2017) [1].
Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.
Learning tasks play an increasingly prominent role in quantum information and computation. They range from fundamental problems such as state discrimination and metrology over the framework of quantum probably approximately correct (PAC) learning, to the recently proposed shadow variants of state tomography. However, the many directions of quantum learning theory have so far evolved separately. We propose a general mathematical formalism for describing quantum learning by training on classical-quantum data and then testing how well the learned hypothesis generalizes to new data. In this framework, we prove bounds on the expected generalization error of a quantum learner in terms of classical and quantum information-theoretic quantities measuring how strongly the learner's hypothesis depends on the specific data seen during training. To achieve this, we use tools from quantum optimal transport and quantum concentration inequalities to establish non-commutative versions of decoupling lemmas that underlie recent information-theoretic generalization bounds for classical machine learning. Our framework encompasses and gives intuitively accessible generalization bounds for a variety of quantum learning scenarios such as quantum state discrimination, PAC learning quantum states, quantum parameter estimation, and quantumly PAC learning classical functions. Thereby, our work lays a foundation for a unifying quantum information-theoretic perspective on quantum learning.