Noisy Quantum Learning Theory

We develop a framework for learning from noisy quantum experiments, focusing on fault-tolerant devices accessing uncharacterized systems through noisy couplings. Our starting point is the complexity class $\textsf{NBQP}$ ("noisy BQP"), modeling noisy fault-tolerant quantum computers that cannot, in general, error-correct the oracle systems they query. Using this class, we show that for natural oracle problems, noise can eliminate exponential quantum learning advantages of ideal noiseless learners while preserving a superpolynomial gap between NISQ and fault-tolerant devices. Beyond oracle separations, we study concrete noisy learning tasks. For purity testing, the exponential two-copy advantage collapses under a single application of local depolarizing noise. Nevertheless, we identify a setting motivated by AdS/CFT in which noise-resilient structure restores a quantum learning advantage in a noisy regime. We then analyze noisy Pauli shadow tomography, deriving lower bounds that characterize how instance size, quantum memory, and noise control sample complexity, and design algorithms with parametrically similar scalings. Together, our results show that the Bell-basis and SWAP-test primitives underlying most exponential quantum learning advantages are fundamentally fragile to noise unless the experimental system has latent noise-robust structure. Thus, realizing meaningful quantum advantages in future experiments will require understanding how noise-robust physical properties interface with available algorithmic techniques.

Learning quantum states and unitaries of bounded gate complexity

While quantum state tomography is notoriously hard, most states hold little interest to practically-minded tomographers. Given that states and unitaries appearing in Nature are of bounded gate complexity, it is natural to ask if efficient learning becomes possible. In this work, we prove that to learn a state generated by a quantum circuit with $G$ two-qubit gates to a small trace distance, a sample complexity scaling linearly in $G$ is necessary and sufficient. We also prove that the optimal query complexity to learn a unitary generated by $G$ gates to a small average-case error scales linearly in $G$. While sample-efficient learning can be achieved, we show that under reasonable cryptographic conjectures, the computational complexity for learning states and unitaries of gate complexity $G$ must scale exponentially in $G$. We illustrate how these results establish fundamental limitations on the expressivity of quantum machine learning models and provide new perspectives on no-free-lunch theorems in unitary learning. Together, our results answer how the complexity of learning quantum states and unitaries relate to the complexity of creating these states and unitaries.