1st International Workshop on
Quantum Software and Quantum Machine Learning (QSML)
UTS: Centre for Quantum Software and Information
Ronald de Wolf | CWI, University of Amsterdam
Title: Quantum Learning Theory
Hao-Chung Cheng, MH, Ping-Cheng Yeh. The learnability of unknown quantum measurements. QIC 16(7&8):615–656 (2016).
Unknown Function
Training Data
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Learning
Algorithm
Comp. Complexity
Sample Complexity
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Photo Credit: Akram Youssry
Training Data
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[1] Vapnik, Springer-Verlag, New York/Berlin, 1982.
[2] Blumer, Ehrenfeucht, Haussler, and Warmuth, Assoc. Comput. Machine, vol. 36, no. 4, pp. 151--160, 1989.
mH=ϵ2C(fatH(8ϵ)⋅log(ϵ2)+log(8/δ))
[1] Bartlett, Long, and Williamson, J. Comput. System Sci., vol. 52, no. 3, pp. 434--452, 1996.
[2] Alon, Ben-David, Cesa-Bianchi, and Haussler, J. ACM, vol. 44, no. 4, pp. 616--631, 1997.
[3] Mendelson, Inventiones Mathematicae, vol. 152, pp. 37--55, 2003.
Hypothesis Set
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S1d and S∞d are polar to each other.
S={x1,…,xn}⊂BX is ϵ-shattered by BX∗ if, for a1,…,an∈R,
ϵi=1∑n∣ai∣≤i=1∑naixiX,
[1] Mendelson and Schechtman, The Shattering Dimension of Sets of Linear Functionals, The Annals of Probability, 32 (3A): 1746–1770, 2004
S={x1,…,xn}⊂BX is ϵ-shattered by BX∗ if, for a1,…,an∈R,
ϵi=1∑n∣ai∣≤i=1∑naixiX,
Joel A. Tropp, Foundations of Computational Mathematics, 12 (4): 389–434, 2011.
[Noncommutative Khintchine inequalities]