Learning Quantum Objects
1st International Workshop on
Quantum Software and Quantum Machine Learning (QSML)
UTS: Centre for Quantum Software and Information
Title Text
This talk concerns
QIP 2018 Tutorial
Ronald de Wolf | CWI, University of Amsterdam
Title: Quantum Learning Theory
- Complexity of Learning
- Full Quantum Settings
Hao-Chung Cheng, MH, Ping-Cheng Yeh. The learnability of unknown quantum measurements. QIC 16(7&8):615–656 (2016).
Unknown Function
Training Data
Hypothesis Set
Learning
Algorithm
Comp. Complexity
Sample Complexity
Hypothesis Set
Training Data
Unknown Function
Photo Credit: Akram Youssry
Training Data
Hypothesis Set
Unknown Function
Given a loss function
find
where
Empirical Risk Minimization
Out-of-Sample Error
In-Sample Error
Training Data
Hypothesis Set
Unknown Function
if for any \(\epsilon>0\)
Probably Approximately Correct (PAC) Learnable
\(\mathcal{H}\) is PAC learnable
$$ \lim_{n\to\infty}\sup_{\mu} \Pr\{\sup_{h\in\mathcal{H}}|R(h) - R_n(h)| >\epsilon\} = 0$$
Sample Complexity
\(\sup_{\mu} \Pr \left\{ \sup_{h\in\mathcal{H}} \big|R(h)-R_n(h)\big|\geq \epsilon \right\}\leq \delta\)
Sample complexity \(m_\mathcal{H}(\epsilon,\delta)\) is the first quantity such that
for every \(n\geq m_\mathcal{H}(\epsilon,\delta),\)
\(m_{\mathcal{H}}(\epsilon,\delta)= \frac{C}{\epsilon^2}\left(\text{VCdim}(\mathcal{H})\log\left(\frac{2}{\epsilon}\right)+\log\left(\frac{2}{\delta}\right)\right)\)
For Boolean functions \(\mathcal{H}\)
[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.
\(\mathcal{X}=\mathbb{R}^2\), \(\mathcal{H}=\{f:\mathcal{X}\to\{0,1\},linear\}\)
VC Dimension
For Real functions \(\mathcal{H}\)
fat\(_\mathcal{H}(\epsilon,\mathcal{X})=\sup\{|\mathcal{S}|: \mathcal{S}\) is \(\epsilon\)-shattered by \(\mathcal{H}\}\)
\(\mathcal{H}\) \(\epsilon\)-Shatters \(\mathcal{S}=\{x_1,\cdots,x_n\}\) if
For Real functions \(\mathcal{H}\)
\(m_{\mathcal{H}}= \frac{C}{\epsilon^2}\left(\text{fat}_{\mathcal{H}}(\frac{\epsilon}{8})\cdot\log(\frac2\epsilon)+\log(8/\delta)\right)\)
[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.
Sample Complexity for Learning Quantum Objects
Q. State
Measurement
Hypothesis Set
Training Data
Unknown Function
Hypothesis Set
Training Data
Unknown Function
Learning Unknown Measurement
Hypothesis Set
Training Data
Unknown Function
Learning States
Learning Measurements
Hypothesis Set
Training Data
Unknown Function
Hypothesis Set
Training Data
Unknown Function
fat\(_{\mathcal{D}(\mathcal{H})}(\epsilon,\mathcal{E}(\mathcal{H})) = O(\log d/\epsilon^2)\)
Sample Complexity for Learning Quantum States
What is the sample complexity of learning unknown measurements?
Learning States
Learning Measurements
fat\(_{\mathcal{D}(\mathcal{H})}(\epsilon,\mathcal{E}(\mathcal{H})) = O(\log d/\epsilon^2)\)
fat\(_{\mathcal{E}(\mathcal{H})}(\epsilon,\mathcal{D}(\mathcal{H})) = O( d/\epsilon^2)\)
Technical Merits
-
The two problems can be solved in the same way.
-
You don't need to know quantum mechanics.
\(S_1^d\) and \(S_\infty^d\) are polar to each other.
State - Measurement Duality
\(\mathcal{S}=\{x_1,\ldots,x_n\}\subset B_X\) is \(\epsilon\)-shattered by \(B_{X^*}\) if, for \(a_1,\ldots,a_n\in\mathbb{R}\),
$$\epsilon\sum_{i=1}^n|a_i|\leq \left\|\sum_{i=1}^n a_i x_i\right\|_\mathcal{X},$$
Choose \(\{a_i\}\) to be independent and uniform \(\{+1,-1\}\) RVs.
LHS = \(\epsilon n\)
[1] Mendelson and Schechtman, The Shattering Dimension of Sets of Linear Functionals, The Annals of Probability, 32 (3A): 1746–1770, 2004
Find \(C(n,d)\) that upper bounds \(\mathbb{E}\left\|\sum_{i=1}^n a_i x_i\right\|_{\mathcal{X}}\)
\(\epsilon n \leq C(n,d)\)
\(\mathcal{S}=\{x_1,\ldots,x_n\}\subset B_X\) is \(\epsilon\)-shattered by \(B_{X^*}\) if, for \(a_1,\ldots,a_n\in\mathbb{R}\),
$$\epsilon\sum_{i=1}^n|a_i|\leq \left\|\sum_{i=1}^n a_i x_i\right\|_\mathcal{X},$$
Learning Q. States
\(B_X:=S_\infty^d\) and \(B_{X^*}:= S_1^d\)
\(\mathbb{E}\left\|\sum_{i=1}^n a_i x_i\right\|_{\infty}\)
\(\leq \sqrt{2\sigma^2 \log d}\)
Joel A. Tropp, Foundations of Computational Mathematics, 12 (4): 389–434, 2011.
\(\sigma^2=\left\|\mathbb{E}\left(\sum_{i=1}^n a_i x_i\right)^2\right\|_\infty\leq n\)
\(\epsilon n \leq \sqrt{2n\log d}\)
Learning Measurement
\(B_X:=S_1^d\) and \(B_{X^*}:= S_\infty^d\)
\(\mathbb{E}\left\|\sum_{i=1}^n a_i x_i\right\|_{1}\)
\(\leq \sqrt{n d}\)
[Noncommutative Khintchine inequalities]