Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions
Fuente:
arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914740166459392 |
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| author | Rouzé, Cambyse Wirth, Melchior Zhang, Haonan |
| author_facet | Rouzé, Cambyse Wirth, Melchior Zhang, Haonan |
| contents | We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_07279 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions Rouzé, Cambyse Wirth, Melchior Zhang, Haonan Functional Analysis Information Theory Mathematical Physics Quantum Physics We extend three related results from the analysis of influences of Boolean functions to the quantum setting, namely the KKL Theorem, Friedgut's Junta Theorem and Talagrand's variance inequality for geometric influences. Our results are derived by a joint use of recently studied hypercontractivity and gradient estimates. These generic tools also allow us to derive generalizations of these results in a general von Neumann algebraic setting beyond the case of the quantum hypercube, including examples in infinite dimensions relevant to quantum information theory such as continuous variables quantum systems. Finally, we comment on the implications of our results as regards to noncommutative extensions of isoperimetric type inequalities, quantum circuit complexity lower bounds and the learnability of quantum observables. |
| title | Quantum Talagrand, KKL and Friedgut's theorems and the learnability of quantum Boolean functions |
| topic | Functional Analysis Information Theory Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2209.07279 |