Berlin 2024 – scientific programme

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QI: Fachverband Quanteninformation

QI 9: Quantum Machine Learning and Classical Simulability

QI 9.7: Talk

Tuesday, March 19, 2024, 11:30–11:45, HFT-FT 101

Exponential concentration in quantum kernel methods — •Supanut Thanasilp¹, Samson Wang², Marco Cerezo³, and Zoe Holmes¹ — ¹EPFL, Lausanne, Switzerland — ²Imperial college London, London, UK — ³Los Alamos National Laboratory, New Mexico, US

Kernel methods in Quantum Machine Learning have recently gained significant attention as a candidate for achieving a quantum advantage. Among attractive properties, when training a kernel-based model one is guaranteed to find the optimal models parameters due to the convexity of the landscape. However, this is based on the assumption that the kernel can be efficiently obtained from quantum hardware. In this work we study the performance of quantum kernel models from the perspective of the resources needed to accurately estimate kernel values. We show that, under certain conditions, values of quantum kernels over different input data can be exponentially concentrated (in the number of qubits) towards some fixed value. Thus on training with a polynomial number of measurements, one ends up with a trivial model where the predictions on unseen inputs are independent of the training data. We identify four sources that can lead to concentration including expressivity of data embedding, global measurements, entanglement and noise. For each source, an associated concentration bound of quantum kernels is analytically derived. Lastly, we show that when dealing with classical data, training a parametrized data embedding with a kernel alignment method is also susceptible to exponential concentration.

Keywords: quantum kernel methods; exponential concentration; barren plateaus; quantum feature map; quantum embedding