# Machine Learning Meets Quantum Computation

### University of Technology Sydney

f: X\to Y

Unknown Function

\{(x_i,y_i)\}_{i=1}^N

Training Data

\mathcal{H}

Hypothesis Set

Learning

Algorithm

\hat{f}

Comp. Complexity

Sample Complexity

# Quantum Challenge #1

## Moment Generating Function: $$\mathbb{E}e^{\theta (A+B)}\neq\mathbb{E}e^{\theta A}e^{\theta B}$$

\frac{a}{b} \mapsto A B^{-1}?
e^{a+b} \mapsto e^A e^B?

=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}

Q
R
S
T

# Classical Mechanics

## $$\theta=(Q+R)S+(R-Q)T\leq 2$$

### Let  $$\text{p}(qrst) := \text{Pr}\{Q=q,R=r,S=s,T=t\}$$.

\mathbb{E}[\theta]= \sum_{qrst}\text{p} (qrst)(qs+rs+rt-qt)
\leq 2

# Quantum Mechanics

|\Psi_{AB}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A|1\rangle_B -|1\rangle_A |0\rangle_B\right)
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
Q
R
S
T
Q=Z
R=X
S=\frac{-Z-X}{\sqrt{2}}
T=\frac{Z-X}{\sqrt{2}}

# Quantum Mechanics

\mathbb{E}[\theta] = \langle QS\rangle + \langle RS\rangle + \langle RT\rangle - \langle QT\rangle= 2\sqrt{2}

# Why Quantum Computation Matters?

Many More!

### Type of Algorithms

CQ
CC
QC
QQ
CQ
QQ
QC

• ### Quantum Control

CC

# Discrete Fourier Transform

y_k = \frac{1}{\sqrt{N}}\sum_{\ell=0}^{N-1} e^{i\frac{2\pi}{N} \ell k} x_\ell

# Quantum Fourier Transform requires $$\Theta (n^2)$$ operations if $$N=2^n$$.

CQ
QQ
CC

# Sample Complexity

\{(x_i,y_i)\}_{i=1}^N

Training Data

R_n(h) = \frac{1}{N}\sum_{i=1}^N \ell (h(x_i), y_i)
h\in\mathcal{H}

Hypothesis Set

f: X\to Y

Unknown Function

## Given a loss function

\ell:Y\times Y \to \mathbb{R}

## find

{f}_n = \arg \min_{h\in \mathcal{H}} R_n (h)

# Sample Complexity

## For Boolean functions $$\mathcal{H}$$

 Vapnik, Springer-Verlag, New York/Berlin, 1982.

 Blumer, Ehrenfeucht, Haussler, and Warmuth, Assoc. Comput. Machine, vol. 36, no. 4, pp. 151--160, 1989.

## There are only very limited matrix concentration results!!

 Joel Tropp. User-friendly tail bounds for sums of random matrices. arXiv:1004.4389.

# Sample Complexity for Learning Quantum Objects

## Measurement

f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
f_\rho(E) = \text{Tr} E\rho

Hypothesis Set

\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
\{(E_i,f_\rho(E_i)\}_{i=1}^N

Training Data

Unknown Function

f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}

Hypothesis Set

\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
\{(E_i,f_\rho(E_i)\}_{i=1}^N

Training Data

Unknown Function

f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}

# Sample Complexity for Learning Quantum States

f_E: \mathcal{D}(\mathcal{H}) \to \mathbb{R}
f_E(\rho) = \text{Tr} E\rho

Hypothesis Set

\{f_{E}:E\in \mathcal{E}(\mathcal{H})\}
\{(\rho_i,f_E(\rho_i)\}_{i=1}^N

Training Data

Unknown Function

f_E : \mathcal{D}(\mathcal{H}) \to \mathbb{R}

# Learning Measurements

## fat$$_{\mathcal{E}(\mathcal{H})}(\epsilon,\mathcal{D}(\mathcal{H})) = O( d/\epsilon^2)$$

Hao-Chung Cheng, MH, Ping-Cheng Yeh. The learnability of unknown quantum measurements. QIC 16(7&8):615–656 (2016).