Reliable quantum supervised learning of a multivariate function mapping depends on the expressivity of the corresponding quantum circuit and measurement resources. We introduce exponential-data-encoding strategies that are optimal amongst all non-entangling Pauli-encoded schemes, which is sufficient for a quantum circuit to express general functions described by Fourier series of very broad frequency spectra using only exponentially few qubits. We show that such an encoding strategy not only mitigates the barren-plateau problem as the function degree scales up, but also exhibits a quantum advantage during loss-function-gradient computation in contrast with known efficient classical strategies when polynomial-depth training circuits are also employed. When gradient-computation resources are constrained, we numerically demonstrate that even exponentialdata-encoding circuits with single-layer training circuits can generally express functions that lie outside the classically-expressible region, thereby supporting the practical benefits of such a quantum advantage.
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