Quantum Learning Boolean Linear Functions w.r.t. Product Distributions
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The problem of learning Boolean linear functions from quantum examples w.r.t. the uniform distribution can be solved on a quantum computer using the Bernstein-Vazirani algorithm. A similar strategy can be applied in the case of noisy quantum training data, as was observed in arXiv:1702.08255v2 [quant-ph]. We employ the biased quantum Fourier transform introduced in arXiv:1802.05690v2 [quant-ph] to develop quantum algorithms for learning Boolean linear functions from quantum examples w.r.t. a biased product distribution. Here, one procedure is applicable to any (except full) bias, the other gives a better performance but is applicable only for small bias. Moreover, we discuss the stability of the second procedure w.r.t. noisy training data and w.r.t. faulty quantum gates. The latter also enables us to solve a version of the problem where the underlying distribution is not known in advance. Finally, we prove lower bounds on the classical and quantum sample complexities of the learning problem and compare these to the upper bounds implied by our algorithms.
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