Quantum learning from remotely accessed quantum compute and data must address two key challenges: verifying the correctness of data and ensuring the privacy of the learner's data-collection strategies and resulting conclusions. The covert (verifiable) learning model of Canetti and Karchmer (TCC 2021) provides a framework for endowing classical learning algorithms with such guarantees. In this work, we propose models of covert verifiable learning in quantum learning theory and realize them without computational hardness assumptions for remote data access scenarios motivated by established quantum data advantages. We consider two privacy notions: (i) strategy-covertness, where the eavesdropper does not gain information about the learner's strategy; and (ii) target-covertness, where the eavesdropper does not gain information about the unknown object being learned. We show: Strategy-covert algorithms for making quantum statistical queries via classical shadows; Target-covert algorithms for learning quadratic functions from public quantum examples and private quantum statistical queries, for Pauli shadow tomography and stabilizer state learning from public multi-copy and private single-copy quantum measurements, and for solving Forrelation and Simon's problem from public quantum queries and private classical queries, where the adversary is a unidirectional or i.i.d. ancilla-free eavesdropper. The lattermost results in particular establish that the exponential separation between classical and quantum queries for Forrelation and Simon's problem survives under covertness constraints. Along the way, we design covert verifiable protocols for quantum data acquisition from public quantum queries which may be of independent interest. Overall, our models and corresponding algorithms demonstrate that quantum advantages are privately and verifiably achievable even with untrusted, remote data.

We study quantum sparse recovery in non-orthogonal, overcomplete dictionaries: given coherent quantum access to a state and a dictionary of vectors, the goal is to reconstruct the state up to $\ell_2$ error using as few vectors as possible. We first show that the general recovery problem is NP-hard, ruling out efficient exact algorithms in full generality. To overcome this, we introduce Quantum Orthogonal Matching Pursuit (QOMP), the first quantum analogue of the classical OMP greedy algorithm. QOMP combines quantum subroutines for inner product estimation, maximum finding, and block-encoded projections with an error-resetting design that avoids iteration-to-iteration error accumulation. Under standard mutual incoherence and well-conditioned sparsity assumptions, QOMP provably recovers the exact support of a $K$-sparse state in polynomial time. As an application, we give the first framework for sparse quantum tomography with non-orthogonal dictionaries in $\ell_2$ norm, achieving query complexity $\widetilde{O}(\sqrt{N}/ε)$ in favorable regimes and reducing tomography to estimating only $K$ coefficients instead of $N$ amplitudes. In particular, for pure-state tomography with $m=O(N)$ dictionary vectors and sparsity $K=\widetilde{O}(1)$ on a well-conditioned subdictionary, this circumvents the $\widetildeΩ(N/ε)$ lower bound that holds in the dense, orthonormal-dictionary setting, without contradiction, by leveraging sparsity together with non-orthogonality. Beyond tomography, we analyze QOMP in the QRAM model, where it yields polynomial speedups over classical OMP implementations, and provide a quantum algorithm to estimate the mutual incoherence of a dictionary of $m$ vectors in $O(m/ε)$ queries, improving over both deterministic and quantum-inspired classical methods.

But nonetheless, they tend to perform well on unseen test data.

But nonetheless, they tend to perform well on unseen test data.

We show that the perturbation set and perturbation function models are equivalent. U ( x): = { g ( x): g 2G }, which completes the proof of this direction. B.1 Proper -Probabilistically Robust PAC Learning for finite G We show that if G is finite then VC classes are -probabilistically robustly learnable. Since A ( S) 2H, by construction of H, there are at least m points in C where A is not probabilistically robustly correct. Using a variant of Markov's inequality, gives We now use the same reasoning in Montasser et al. [2019], to show that no proper learning rule works.

Recently, Montasser et al. [2019] showed that finite VC dimension is not sufficient

We can only see the exact value realized in a box if we open it and pay the opening cost.