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Many applications of quantum computing in the near term rely on variationalquantum circuits (VQCs). They have been showcased as a promising model forreaching a quantum advantage in machine learning with current noisyintermediate scale quantum computers (NISQ). It is often believed that thepower of VQCs relies on their exponentially large feature space, and extensiveworks have explored the expressiveness and trainability of VQCs in that regard.In our work, we propose a classical sampling method that may closelyapproximate a VQC with Hamiltonian encoding, given only the description of itsarchitecture. It uses the seminal proposal of Random Fourier Features (RFF) andthe fact that VQCs can be seen as large Fourier series. We provide generaltheoretical bounds for classically approximating models built fromexponentially large quantum feature space by sampling a few frequencies tobuild an equivalent low dimensional kernel, and we show experimentally thatthis approximation is efficient for several encoding strategies. Precisely, weshow that the number of required samples grows favorably with the size of thequantum spectrum. This tool therefore questions the hope for quantum advantagefrom VQCs in many cases, but conversely helps to narrow the conditions fortheir potential success. We expect VQCs with various and complex encodingHamiltonians, or with large input dimension, to become more robust to classicalapproximations.

Classically Approximating Variational Quantum Machine Learning with Random Fourier Features