Machine Learning Meets Quantum Computation

Min-Hsiu Hsieh (謝明修)

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 Computation 

Classical Bit \(x\in\mathbb{Z}=\{0,1\}\)

QuBit \(\rho\in\mathbb{C}^{2\times 2}\geq0\) & Tr\([\rho]=1\)

Random Bit \(\left(\begin{array}{cc} p(0) & 0\\ 0 & p(1) \end{array}\right)\) is a special case.

Quantum Computation 

Quantum Operation: \(\rho\mapsto\sigma\) 

Unitary is a special case. 

Quantum Measurement: \(\rho\mapsto\mathbb{R}\) 

Quantum Challenge #1

Noncommutative: \(AB\neq BA\) 

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?

Quantum Challenge #2

Entanglement: \(\rho_{AB}\neq \rho_{A}\otimes\rho_B\) 

Problem Setup

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

Alice

Bob

Compute \((QS+RS+RT-QT)\)

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

Probabilistically, 

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 Input

Type of Algorithms

CQ
CC
QC
QQ
CQ
QQ
QC
  • Linear Equation Solvers

  • Peceptron

  • Recommendation Systems

  • Semidefinite Programming

  • Many Others (such as non-Convex Optimization)

  • State Tomography

  • Entanglement Structure

  • Quantum Control

CC
  • Linear Equation Solvers

  • Recommendation Systems

  • Semidefinite Programming

  • Minimum Conical Hull  

Quantum-Inspired Classical Algorithms 

\(A\mathbf{x} = \mathbf{b}\)

\(A =\sum \sigma_\ell |u_\ell\rangle\langle v_\ell|\)

\(\mathbf{x} =\sum \lambda_\ell |v_\ell\rangle\)

Discrete Fourier Transform

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

Classical Fast Fourier Transform requires \(\Theta (n2^n)\) operations if \(N=2^n\).

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)

where

Empirical Risk Minimization

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.

\(Z=\sup_{f\in\mathcal{F}}\big| \sum_{i=1}^n f(x_i)\big|\)

\({Z}=\sup_{\bm{f}\in\mathcal{F}}\left\| \sum_{i=1}^n \bm{f}(\bm{X}_i)\right\|_p.\)

There are only very limited matrix concentration results!!

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

Sample Complexity for Learning Quantum Objects

Q. State

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}

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

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 Unknown Measurement

Learning States

Learning Measurements

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

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

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

Thank you for your attention!

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