Parameterized Quantum Circuits

Min-Hsiu Hsieh

(UTS)

and Its Applications in Machine Learning

Joint Work with

Yuxuan Du

Tongliang Liu

Dacheng Tao

(USyd)

Machine Learning Tasks

PQC Can Achieve

"In computational complexity-theoretic terms, this generally means providing a superpolynomial speedup over the best possible classical algorithm."

Quantum supremacy

QML applications are natural candidates for NISQ devices.

Part I

Parameterized Quantum

Circuits

and many more!

PQC

q(X)

q(X)

\theta

\theta

Classical Optimization

\theta = \arg \min_{\bm{\theta}} \mathcal{L}(q(X),p(X))

\theta = \arg \min_{\bm{\theta}} \mathcal{L}(q(X),p(X))

Given an loss function \(\mathcal{L}\), PQC output \(q(X)\), a target distribution \(p(X)\),

Part II

PQC as Quantum Perceptron

and more!

Quantum Perceptron

Grover Search

P_\omega= |\omega\rangle\langle \omega|

P_\omega= |\omega\rangle\langle \omega|

P_s= |+\rangle\langle +|^{\otimes n}

P_s= |+\rangle\langle +|^{\otimes n}

U_s= 2|+\rangle\langle +|^{\otimes n} - I

U_s= 2|+\rangle\langle +|^{\otimes n} - I

U_\omega= 2|\omega\rangle\langle \omega| - I

U_\omega= 2|\omega\rangle\langle \omega| - I

How to identify

from

Grover Search

Variational Grover Search

\min_{\phi} \langle \phi |P_{\omega^\perp} | \phi\rangle

\min_{\phi} \langle \phi |P_{\omega^\perp} | \phi\rangle

Learn \(|\phi\rangle\) using trainable quantum circuits.

Morales, Tlyachev, and Biamonte. Variationally Learning Grover’s Quantum Search Algorithm. arXiv:1805.09337

U(\alpha) = e^{i\alpha P_\omega}

U(\alpha) = e^{i\alpha P_\omega}

V(\beta) = e^{i\beta P_s}

V(\beta) = e^{i\beta P_s}

Variational Grover Search

Variational Quantum Perceptron

\mathcal{D} = \{\bm{x}_i \in \mathbb{R}^M, y_i\in\{\pm 1\}\}_{i=1}^N

\mathcal{D} = \{\bm{x}_i \in \mathbb{R}^M, y_i\in\{\pm 1\}\}_{i=1}^N

|\Phi^{(k)}\rangle_{F,I}= U_{data}(\mathcal{D})

|\Phi^{(k)}\rangle_{F,I}= U_{data}(\mathcal{D})

=\frac{1}{\sqrt{N}}\left(\sum_{i=0,i\neq k}^{N-1}|\psi_i^{(0)}\rangle_{F}|i\rangle_I -|\psi_k^{(1)}\rangle_{F}|k\rangle_I\right).

=\frac{1}{\sqrt{N}}\left(\sum_{i=0,i\neq k}^{N-1}|\psi_i^{(0)}\rangle_{F}|i\rangle_I -|\psi_k^{(1)}\rangle_{F}|k\rangle_I\right).

Variational Quantum Perceptron

U_{c1}|\Phi^k\rangle_{F,I}=|\varphi\rangle_F\otimes\left(\cos3\theta|B\rangle_I+\sin3\theta|k\rangle_I\right)

U_{c1}|\Phi^k\rangle_{F,I}=|\varphi\rangle_F\otimes\left(\cos3\theta|B\rangle_I+\sin3\theta|k\rangle_I\right)

U_{L_1}|{\Phi^k}\rangle_{F,I}= \cos\theta(\textcolor{red}{\alpha_B}|\psi_B^{(0)}\rangle_F+ \textcolor{green}{\beta_k}|\psi_k^{(1)}\rangle_F)|{B}\rangle_I + \sin\theta(-\textcolor{red}{\alpha_B}|\psi_B^{(0)}\rangle_F+ \textcolor{green}{\beta_k}|\psi_k^{(1)}\rangle_F)|{k}\rangle_I