The complexity of quantum support vector machines

Gian Gentinetta1,2, Arne Thomsen3,2, David Sutter2, and Stefan Woerner2

1Institute of Physics, École Polytechnique Fédérale de Lausanne
2IBM Quantum, IBM Research Europe – Zurich
3Department of Physics, ETH Zurich

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Abstract

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 $O(M^{4.67}/\varepsilon^2)$ quantum circuit evaluations, where $M$ denotes the size of the data set and $\varepsilon$ the solution accuracy compared to the ideal result from exact expectation values, which is only obtainable in theory. We prove under an empirically motivated assumption that the kernelized primal problem can alternatively be solved in $O(\min \{ M^2/\varepsilon^6, \, 1/\varepsilon^{10} \})$ 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.

Quantum support vector machines are candidates for demonstrating a quantum advantage in the close future for classification problems. The idea is that a (hopefully useful) kernel function is given by an efficient quantum circuit which is classically hard to evaluate. Due to the probabilistic nature of quantum mechanics such kernel functions are affected by statistical uncertainty. In this work, we rigorously analyze how this uncertainty (also called shot noise) impacts the overall computational complexity for solving the classification problem.

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