Exponential concentration in quantum kernel methods
Journal Article
·
· Nature Communications
- National Univ. of Singapore (Singapore); Ecole Polytechnique Federale Lausanne (EPFL) (Switzerland); Chulalongkorn University, Bangkok (Thailand)
- Imperial College, London (United Kingdom)
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States); Oak Ridge National Laboratory (ORNL), Oak Ridge, TN (United States). Quantum Science Center (QSC)
- Ecole Polytechnique Federale Lausanne (EPFL) (Switzerland); Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
Kernel methods in Quantum Machine Learning (QML) have recently gained significant attention as a potential candidate for achieving a quantum advantage in data analysis. Among other attractive properties, when training a kernel-based model one is guaranteed to find the optimal model’s parameters due to the convexity of the training landscape. However, this is based on the assumption that the quantum 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 input 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. Our results are verified through numerical simulations for several QML tasks. Altogether, we provide guidelines indicating that certain features should be avoided to ensure the efficient evaluation of quantum kernels and so the performance of quantum kernel methods.
- Research Organization:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Organization:
- USDOE National Nuclear Security Administration (NNSA)
- Grant/Contract Number:
- 89233218CNA000001
- OSTI ID:
- 2483516
- Alternate ID(s):
- OSTI ID: 2470554
- Report Number(s):
- LA-UR--22-28427
- Journal Information:
- Nature Communications, Journal Name: Nature Communications Journal Issue: 1 Vol. 15; ISSN 2041-1723
- Publisher:
- Nature Publishing GroupCopyright Statement
- Country of Publication:
- United States
- Language:
- English