Summability of Multi-Dimensional Trigonometric Fourier Series

Weisz, Ferenc

doi:10.48550/arxiv.1206.1789

Cited by 7 publications

(10 citation statements)

References 73 publications

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“…Proof. To begin with, we note that we can approximate any given g ∈ L 2 ([0, 2π] N ), up to an arbitrarily small error in L 2 norm, by using a truncated Fourier series [43]. More specifically, for any given > 0, there exists some K ∈ N and some set of coefficients…”

Section: Appendix B: Non-integer Frequenciesmentioning

confidence: 99%

Effect of data encoding on the expressive power of variational quantum-machine-learning models

2021

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Quantum computers can be used for supervised learning by treating parametrised quantum circuits as models that map data inputs to predictions. While a lot of work has been done to investigate practical implications of this approach, many important theoretical properties of these models remain unknown. Here we investigate how the strategy with which data is encoded into the model influences the expressive power of parametrised quantum circuits as function approximators. We show that one can naturally write a quantum model as a partial Fourier series in the data, where the accessible frequencies are determined by the nature of the data encoding gates in the circuit. By repeating simple data encoding gates multiple times, quantum models can access increasingly rich frequency spectra. We show that there exist quantum models which can realise all possible sets of Fourier coefficients, and therefore, if the accessible frequency spectrum is asymptotically rich enough, such models are universal function approximators.

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Section: Appendix B: Non-integer Frequenciesmentioning

confidence: 99%

Effect of data encoding on the expressive power of variational quantum-machine-learning models

2021

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“…Proof First, it is known that any even square-integrable function f (x) can be approximated by a truncated Fourier series [40]. For any > 0, there exists a K ∈ N + such that…”

Section: Proof Of Propositionmentioning

confidence: 99%

Power and limitations of single-qubit native quantum neural networks

Yu¹,

Yao²,

Li³

et al. 2022

Preprint

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Quantum neural networks (QNNs) have emerged as a leading strategy to establish applications in machine learning, chemistry, and optimization. While the applications of QNN have been widely investigated, its theoretical foundation remains less understood. In this paper, we formulate a theoretical framework for the expressive ability of data re-uploading quantum neural networks that consist of interleaved encoding circuit blocks and trainable circuit blocks. First, we prove that single-qubit quantum neural networks can approximate any univariate function by mapping the model to a partial Fourier series. Beyond previous works' understanding of existence, we in particular establish the exact correlations between the parameters of the trainable gates and the working Fourier coefficients, by exploring connections to quantum signal processing. Second, we discuss the limitations of singlequbit native QNNs on approximating multivariate functions by analyzing the frequency spectrum and the flexibility of Fourier coefficients. We further demonstrate the expressivity and limitations of single-qubit native QNNs via numerical experiments. As applications, we introduce natural extensions to multi-qubit quantum neural networks, which exhibit the capability of classifying real-world multi-dimensional data. We believe these results would improve our understanding of QNNs and provide a helpful guideline for designing powerful QNNs for machine learning tasks.

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“…The phase embedding that we investigate was considered in Schuld et al [34] for continuous-valued inputs. They showed that if the phase embedding is repeated r times sequentially or in parallel, then the output of the variational model is expressible as the r th cubic partial sum [36] of a Fourier series:…”

Section: A Phase Embeddingmentioning

confidence: 99%

“…The observable and trainable blocks control the coefficients of the Fourier basis elements in the partial sum. Since every function in L 2 ([0, 2π] n ) can be represented by the limit of a Fourier series [36], VQML models can approximate any function in this space to arbitrarily small error, in L 2 norm, by using an embedding scheme that produces the required Fourier spectrum. This also assumes that the observable and trainable blocks can fit the Fourier coefficients to the desired error.…”

Section: Introductionmentioning

confidence: 99%