Reliable quantum kernel classification using fewer circuit evaluations

Quantum kernel methods are a candidate for quantum speed-ups in supervised machine learning. The number of quantum measurements $N$ required for a reasonable kernel estimate is a critical resource, both from complexity considerations and because of the constraints of near-term quantum hardware. We emphasize that for classification tasks, the aim is accurate classification and not accurate kernel evaluation, and demonstrate that the former is more resource efficient. In general, the uncertainty in the quantum kernel, arising from finite sampling, leads to misclassifications over some kernel instantiations. We introduce a suitable performance metric that characterizes the robustness or reliability of classification over a dataset, and obtain a bound for $N$ which ensures, with high probability, that classification errors over a dataset are bounded by the margin errors of an idealized quantum kernel classifier. Using techniques of robust optimization, we then show that the number of quantum measurements can be significantly reduced by a robust formulation of the original support vector machine. We consider the SWAP test and the GATES test quantum circuits for kernel evaluations, and show that the SWAP test is always less reliable than the GATES test for any $N$. Our strategy is applicable to uncertainty in quantum kernels arising from {\em any} source of noise, although we only consider the statistical sampling noise in our analysis.