Constrained and Vanishing Expressivity of Quantum Fourier Models
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arXiv
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| Autores principales: | , , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866908512959856640 |
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| author | Mhiri, Hela Monbroussou, Leo Herrero-Gonzalez, Mario Thabet, Slimane Kashefi, Elham Landman, Jonas |
| author_facet | Mhiri, Hela Monbroussou, Leo Herrero-Gonzalez, Mario Thabet, Slimane Kashefi, Elham Landman, Jonas |
| contents | In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier Series which frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially when the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for Quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviours are observed in numerical simulations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constrained and Vanishing Expressivity of Quantum Fourier Models Mhiri, Hela Monbroussou, Leo Herrero-Gonzalez, Mario Thabet, Slimane Kashefi, Elham Landman, Jonas Quantum Physics In this work, we highlight an unforeseen behavior of the expressivity of Parameterized Quantum Circuits (PQCs) for machine learning. A large class of these models, seen as Fourier Series which frequencies are derived from the encoding gates, were thought to have their Fourier coefficients mostly determined by the trainable gates. Here, we demonstrate a new correlation between the Fourier coefficients of the quantum model and its encoding gates. In addition, we display a phenomenon of vanishing expressivity in certain settings, where some Fourier coefficients vanish exponentially when the number of qubits grows. These two behaviors imply novel forms of constraints which limit the expressivity of PQCs, and therefore imply a new inductive bias for Quantum models. The key concept in this work is the notion of a frequency redundancy in the Fourier series spectrum, which determines its importance. Those theoretical behaviours are observed in numerical simulations. |
| title | Constrained and Vanishing Expressivity of Quantum Fourier Models |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2403.09417 |