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arXiv:2303.05428 · ACM International Conference on Computing Frontiers (CF)

Quantum splines for non-linear approximations

Antonio Macaluso, L. Clissa, Stefano Lodi, Claudio Sartori

Published May 11, 2020• 8 citations
arXivHTMLPDF

Abstract

Quantum Computing offers a new paradigm for efficient computing and many AI applications could benefit from its potential boost in performance. However, the main limitation is the constraint to linear operations that hampers the representation of complex relationships in data. In this work, we propose an efficient implementation of quantum splines for non-linear approximation. In particular, we first discuss possible parametrisations, and select the most convenient for exploiting the HHL algorithm to obtain the estimates of spline coefficients. Then, we investigate QSpline performance as an evaluation routine for some of the most popular activation functions adopted in ML. Finally, a detailed comparison with classical alternatives to the HHL is also presented.

DOI
10.1145/3387902.3394032

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HHL algorithm \cite{harrow} to solve linear system is a powerful and efficient quantum technique to deal with many matrix operations (such as matrix multiplication, powers and inversion). It inspires many applications in quantum machine learning \cite{biamonte, dunjko}. However, due to the restrictions of HHL algorithm itself, many quantum machine learning algorithms also share one or two restrictions. The most common restrictions include quantum state preparation, condition number and Hamiltonian simulation. In this work, we first give an efficient quantum algorithm to achieve quantum state preparation, which actually achieves an exponential speedup than the algorithms given in \cite{clader,lloyd13}. Then we provide an application of HHL algorithm in cubic spline interpolation problem. We will show that in this problem, the condition number is small, the preparation of quantum state is efficient based on the new algorithm we proposed and the Hamiltonian simulation is efficiently implemented. So the quantum algorithm obtained by HHL algorithm towards this problem actually achieves an exponential speedup than any classical algorithm with no restrictions. This can be viewed as another application of HHL algorithm with no restrictions after the work of Clader et al \cite{clader} in studying electromagnetic scattering cross-section.
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This paper presents a novel method for non-linear matrix embedding utilizing quantum tunneling. Traditional matrix embedding methods often struggle with representing complex relationships and inherent nonlinearity, leading to limited computational efficiency and potential inaccuracies. This research introduces a technique that leverages quantum tunneling to create a more robust and complex representation of the matrix, offering a potential paradigm shift in data representation and enabling novel applications within quantum machine learning. The core of this approach involves a carefully designed process that exploits the quantum properties of tunneling to enhance the representation of matrix elements, thereby improving the accuracy and efficiency of computations. We demonstrate the efficacy of this method through theoretical analysis and preliminary experimental validation, highlighting its potential for significant advancements across various fields.
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