The complexity of quantum support vector machines
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Quantum support vector machines employ quantum circuits to define the kernel function. It has been shown that this approach offers a provable exponential speedup compared to any known classical algorithm for certain data sets. The training of such models corresponds to solving a convex optimization problem either via its primal or dual formulation. Due to the probabilistic nature of quantum mechanics, the training algorithms are affected by statistical uncertainty, which has a major impact on their complexity. We show that the dual problem can be solved in 𝒪(M^4.67/ε^2) quantum circuit evaluations, where M denotes the size of the data set and ε the solution accuracy. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in 𝒪(min{ M^2/ε^6, 1/ε^10}) evaluations by employing a generalization of a known classical algorithm called Pegasos. Accompanying empirical results demonstrate these analytical complexities to be essentially tight. In addition, we investigate a variational approximation to quantum support vector machines and show that their heuristic training achieves considerably better scaling in our experiments.
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