🤖 AI Summary
This paper investigates the problem of learning unitary operators in resource-constrained quantum systems, where the learner only accesses noisy measurement expectations via quantum statistical queries (QSQs). The core method maps unitary operators to their Choi–Jamiołkowski states, integrates Pauli Fourier analysis with locally scrambled ensembles, and—crucially—adapts the Goldreich–Levin algorithm to the QSQ model for the first time. Key contributions include: (1) polynomial sample complexity for learning log n-juntas, low-influence quantum Boolean functions, and constant-depth quantum circuits; (2) a novel definition of average-case proxy models; (3) exponential and doubly exponential lower bounds for phase oracle learning and channel unitarity testing; and (4) a proof of learnability of unitary operators over locally scrambled ensembles. Collectively, these results establish a rigorous theoretical foundation and an efficient algorithmic framework for quantum machine learning under realistic noise constraints.
📝 Abstract
We propose several algorithms for learning unitary operators from quantum statistical queries (QSQs) with respect to their Choi-Jamiolkowski state. Quantum statistical queries capture the capabilities of a learner with limited quantum resources, which receives as input only noisy estimates of expected values of measurements. Our methods hinge on a novel technique for estimating the Fourier mass of a unitary on a subset of Pauli strings with a single quantum statistical query, generalizing a previous result for uniform quantum examples. Exploiting this insight, we show that the quantum Goldreich-Levin algorithm can be implemented with quantum statistical queries, whereas the prior version of the algorithm involves oracle access to the unitary and its inverse. Moreover, we prove that $mathcal{O}(log n)$-juntas and quantum Boolean functions with constant total influence are efficiently learnable in our model, and constant-depth circuits are learnable sample-efficiently with quantum statistical queries. On the other hand, all previous algorithms for these tasks require direct access to the Choi-Jamiolkowski state or oracle access to the unitary. In addition, our upper bounds imply that the actions of those classes of unitaries on locally scrambled ensembles can be efficiently learned. We also demonstrate that, despite these positive results, quantum statistical queries lead to an exponentially larger sample complexity for certain tasks, compared to separable measurements to the Choi-Jamiolkowski state. In particular, we show an exponential lower bound for learning a class of phase-oracle unitaries and a double exponential lower bound for testing the unitarity of channels, adapting to our setting previous arguments for quantum states. Finally, we propose a new definition of average-case surrogate models, showing a potential application of our results to hybrid quantum machine learning.
