Journal Article10.1038/s41467-024-49287-w
Exponential concentration in quantum kernel methods
Supanut Thanasilp,Samson Wang,M. Cerezo,Zoë Holmes +3 more
TL;DR: Quantum kernel methods can exhibit exponential concentration of values towards a fixed value, leading to trivial models.
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Abstract: Abstract 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.
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Figures
Fig. 9 | Kernel target alignment. The variance of TA(θ) with respect to variational parameters is plotted as a function of n. Here we use the hypercube dataset with Ns = 10. 
Fig. 4 | Hardware Efficient Embedding (HEE). A layer is composed of single qubit x-rotations where the rotation angle on qubit k is given by the kth component of the input data point x. After each layer of rotations, one applies entangling gates acting on adjacent pairs of qubits. 
Fig. 5 | Datasets. a An input data point x is obtained from dimensionally reducing an original MNIST image to n features using principal component analysis. We assign label − 1 if the original image is digit ‘0’ and 1 if the original image is digit ‘1’. b A hypercube of width 2π/21/n is centered at the origin. An input data point x with each of its component bounded between −π and π has an associated label y = 1 if the point is inside the hypercube (represented by a circle) and y = − 1, otherwise (represented by a cross). 
Fig. 6 | Effect of expressivity on quantum kernels. We plot variances of the (a) fidelity and (b) projected quantum kernels, as a function of n and L. The classical data from the MNIST dataset (Ns = 40) is encoded via an L-layer HEE. 
Fig. 8 | Effect of noise.Weplot the average of the difference between the quantum kernels and their respective fixed point μ over different input data points and for different number of layers L and noise parameter q. We consider the fidelity quantum kernel in a with μ = 1/2n and the projected quantum kernel in b with μ = 1. We use the MNIST dataset with Ns = 40 and n = 8. 
Table 1 | Summary of our main results
Citations
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Quantum machine learning beyond kernel methods
Sofiene Jerbi,Lukas J. Fiderer,Hendrik Poulsen Nautrup,Jonas M. Kübler,Hans J. Briegel,Vedran Dunjko +5 more
TL;DR: In this article, the authors extend the applicability of this result to a more general family of parametrized quantum circuit models called data re-uploading circuits and show that models defined and trained variationally can exhibit a critically better generalization performance than their kernel formulations.
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A comprehensive review of quantum machine learning: from NISQ to fault tolerance
Yunfei Wang,Junyu Liu +1 more
TL;DR: This comprehensive review of quantum machine learning covers NISQ technologies, fault-tolerant algorithms, and statistical learning theory, providing an unbiased overview of the field's fundamental concepts, techniques, and applications in both NISQ and fault-tolerant quantum computing.
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The power of one clean qubit in supervised machine learning
M. Karimi,Ali Javadi-Abhari,Christoph Simon,Roohollah Ghobadi +3 more
TL;DR: The DQC1 model is effective for estimating complex kernel functions and implementing binary classification problems. Quantum coherence and discord are leveraged to achieve improved performance compared to traditional methods.
Coreset Selection Can Accelerate Quantum Machine Learning Models with Provable Generalization
Yiming Huang,Xiao Yuan,Huiyuan Wang,Yuxuan Du +3 more
TL;DR: Researchers propose coreset selection to accelerate quantum machine learning models, such as QNNs and quantum kernels, by distilling a subset from the original dataset, achieving comparable performance with reduced training cost and provable generalization.
Effect of alternating layered <i>Ansäatze</i> on trainability of projected quantum kernels
Yudai Suzuki,Muyuan Li +1 more
References
Quantum machine learning
Jacob Biamonte,Jacob Biamonte,Peter Wittek,Nicola Pancotti,Patrick Rebentrost,Nathan Wiebe,Seth Lloyd +6 more
TL;DR: The field of quantum machine learning explores how to devise and implement quantum software that could enable machine learning that is faster than that of classical computers.
Supervised learning with quantum-enhanced feature spaces.
Vojtěch Havlíček,Vojtěch Havlíček,Antonio Corcoles,Kristan Temme,Aram W. Harrow,Abhinav Kandala,Jerry M. Chow,Jay M. Gambetta +7 more
TL;DR: In this article, two quantum algorithms for machine learning on a superconducting processor are proposed and experimentally implemented, using a variational quantum circuit to classify the data in a way similar to the method of conventional SVMs.
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Error Mitigation for Short-Depth Quantum Circuits
TL;DR: Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits by resampling randomized circuits according to a quasiprobability distribution.
Error mitigation extends the computational reach of a noisy quantum processor
Abhinav Kandala,Kristan Temme,Antonio Corcoles,Antonio Mezzacapo,Jerry M. Chow,Jay M. Gambetta +5 more
TL;DR: This work applies the error mitigation protocol to mitigate errors in canonical single- and two-qubit experiments and extends its application to the variational optimization of Hamiltonians for quantum chemistry and magnetism.
Barren plateaus in quantum neural network training landscapes
TL;DR: It is shown that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits.