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Quantum computers are known to provide speedups over classicalstate-of-the-art machine learning methods in some specialized settings. Forexample, quantum kernel methods have been shown to provide an exponentialspeedup on a learning version of the discrete logarithm problem. Understandingthe generalization of quantum models is essential to realizing similar speedupson problems of practical interest. Recent results demonstrate thatgeneralization is hindered by the exponential size of the quantum featurespace. Although these results suggest that quantum models cannot generalizewhen the number of qubits is large, in this paper we show that these resultsrely on overly restrictive assumptions. We consider a wider class of models byvarying a hyperparameter that we call quantum kernel bandwidth. We analyze thelarge-qubit limit and provide explicit formulas for the generalization of aquantum model that can be solved in closed form. Specifically, we show thatchanging the value of the bandwidth can take a model from provably not beingable to generalize to any target function to good generalization forwell-aligned targets. Our analysis shows how the bandwidth controls thespectrum of the kernel integral operator and thereby the inductive bias of themodel. We demonstrate empirically that our theory correctly predicts howvarying the bandwidth affects generalization of quantum models on challengingdatasets, including those far outside our theoretical assumptions. We discussthe implications of our results for quantum advantage in machine learning.

Bandwidth Enables Generalization in Quantum Kernel Models