Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866909574340018176 |
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| author | Gonon, Lukas Jacquier, Antoine |
| author_facet | Gonon, Lukas Jacquier, Antoine |
| contents | Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterised quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomised quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with $\mathcal{O}(\varepsilon^{-2})$ weights and $\mathcal{O} (\lceil \log_2(\varepsilon^{-1}) \rceil)$ qubits suffices to achieve accuracy $\varepsilon>0$ when approximating functions with integrable Fourier transform. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_12904 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs Gonon, Lukas Jacquier, Antoine Quantum Physics Machine Learning Probability 68Q12, 68T07, 65D15 Universal approximation theorems are the foundations of classical neural networks, providing theoretical guarantees that the latter are able to approximate maps of interest. Recent results have shown that this can also be achieved in a quantum setting, whereby classical functions can be approximated by parameterised quantum circuits. We provide here precise error bounds for specific classes of functions and extend these results to the interesting new setup of randomised quantum circuits, mimicking classical reservoir neural networks. Our results show in particular that a quantum neural network with $\mathcal{O}(\varepsilon^{-2})$ weights and $\mathcal{O} (\lceil \log_2(\varepsilon^{-1}) \rceil)$ qubits suffices to achieve accuracy $\varepsilon>0$ when approximating functions with integrable Fourier transform. |
| title | Universal Approximation Theorem and error bounds for quantum neural networks and quantum reservoirs |
| topic | Quantum Physics Machine Learning Probability 68Q12, 68T07, 65D15 |
| url | https://arxiv.org/abs/2307.12904 |