Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions
Rouzé C, Wirth M, Zhang H. 2024. Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions. Communications in Mathematical Physics. 405(4), 95.
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Abstract
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.
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2024-04-09
Journal Title
Communications in Mathematical Physics
Acknowledgement
Open access funding provided by the Carolinas Consortium.
H.Z. is supported by the Lise Meitner fellowship, Austrian Science Fund (FWF) M3337. H.Z. would like to thank the American Institute of Mathematics and the AIM workshop Analysis on the hypercube with applications to quantum computing. He is also grateful to the organizers and other participants for creating an active atmosphere. The research of C.R. has been supported by ANR project QTraj (ANR-20-CE40-0024-01) of the French National Research Agency (ANR). C.R. acknowledges the support of the Munich Center for Quantum Sciences and Technology, as well as the Humboldt Foundation. C.R. would like to thank Amanda Young for fruitful discussion on the applications of Friedgut’s Junta theorem to learning quantum dynamics. The research of M.W. was funded by the Austrian Science Fund (FWF) under the Esprit Programme [ESP 156]. For the purpose of Open Access, the authors have applied a CC BY public copyright licence to any Author Accepted Manuscript (AAM) version arising from this submission. The authors want to thank Francisco Escudero Gutierrez and Hsin-Yuan Huang for helpful comments on an earlier version of the paper. They are grateful to the referees for the careful reading and helpful comments.
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405
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4
Article Number
95
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IST-REx-ID
Cite this
Rouzé C, Wirth M, Zhang H. Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions. Communications in Mathematical Physics. 2024;405(4). doi:10.1007/s00220-024-04981-0
Rouzé, C., Wirth, M., & Zhang, H. (2024). Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions. Communications in Mathematical Physics. Springer Nature. https://doi.org/10.1007/s00220-024-04981-0
Rouzé, Cambyse, Melchior Wirth, and Haonan Zhang. “Quantum Talagrand, KKL and Friedgut’s Theorems and the Learnability of Quantum Boolean Functions.” Communications in Mathematical Physics. Springer Nature, 2024. https://doi.org/10.1007/s00220-024-04981-0.
C. Rouzé, M. Wirth, and H. Zhang, “Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions,” Communications in Mathematical Physics, vol. 405, no. 4. Springer Nature, 2024.
Rouzé C, Wirth M, Zhang H. 2024. Quantum Talagrand, KKL and Friedgut’s theorems and the learnability of quantum boolean functions. Communications in Mathematical Physics. 405(4), 95.
Rouzé, Cambyse, et al. “Quantum Talagrand, KKL and Friedgut’s Theorems and the Learnability of Quantum Boolean Functions.” Communications in Mathematical Physics, vol. 405, no. 4, 95, Springer Nature, 2024, doi:10.1007/s00220-024-04981-0.
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