Recent Breakthrough in Quantum Machine Learning

Min-Hsiu Hsieh

University of Technology Sydney

Biamonte et. al., Nature volume 549, pages 195–202, 2017

Quantum Machine Learning

Quantum Fourier Transform

QFT: \(|x\rangle\mapsto \frac{1}{\sqrt{N}}\sum_{k=1}^N \omega_n^{xk} |x\rangle\), \(N=2^n\)

QFT requires only \(O(n^2)\) gates.

Used in Quantum Phase Estimation, etc.

Quantum Phase Estimation

Given is \(U\) and \(|\psi\rangle\) so that \(U|\psi\rangle = e^{i2\pi\theta}|\psi\rangle\)

QPE can estimate \(\theta\) using \(O(\frac{1}{\epsilon})\) operations.

Used in Shor's and HHL algorithm

Harrow et. al., Phys. Rev. Lett. vol. 15, no. 103, pp. 150502 (2009)

Matrix Inversion

Solving \(A|\bm{x}\rangle=|\bm{b}\rangle\) so that \(\langle\bm{x}|M|\bm{x}\rangle\) can be estimated with \(O(\kappa\log N)\)

The best classical algorithm requires \(O(N\sqrt{\kappa})\) operations.

Grover Search

P_\omega= |\omega\rangle\langle \omega|
Pω=ωωP_\omega= |\omega\rangle\langle \omega|
P_s= |+\rangle\langle +|^{\otimes n}
Ps=++nP_s= |+\rangle\langle +|^{\otimes n}
U_s= 2|+\rangle\langle +|^{\otimes n} - I
Us=2++nIU_s= 2|+\rangle\langle +|^{\otimes n} - I
U_\omega= 2|\omega\rangle\langle \omega| - I
Uω=2ωωIU_\omega= 2|\omega\rangle\langle \omega| - I

How to identify

from

Grover Search

Recommendation Systems

O(\text{poly}(k)\text{poly}\log(mn))
O(poly(k)polylog(mn))O(\text{poly}(k)\text{poly}\log(mn))

Estimate an \(m\times n\) matrix with rank \(k\).

[2] Tang, arXiv:1807.04271, 2018

[1] Kerenidis and Prakash. arXiv:1704.04992, 2017.

Variational Quantum Perceptron and Classification of Nonlinear Data

Du, MH, Liu and  Tao. Implementable Quantum Classifier for Nonlinear Data. arXiv:1809.06056

Variational Grover Search

\min_{\phi} \langle \phi |P_{\omega^\perp} | \phi\rangle
minϕϕPωϕ\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(α)=eiαPωU(\alpha) = e^{i\alpha P_\omega}
V(\beta) = e^{i\beta P_s}
V(β)=eiβPsV(\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
D={xiRM,yi{±1}}i=1N\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})
Φ(k)F,I=Udata(D)|\Phi^{(k)}\rangle_{F,I}= U_{data}(\mathcal{D})
U_k|\Phi^k\rangle_{F,I}=\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).
UkΦkF,I=1N(i=0,ikN1ψi(0)FiIψk(1)FkI).U_k|\Phi^k\rangle_{F,I}=\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_{dis}\otimes\mathbb{I})U_f|\Phi^k\rangle_{F,I}=|\Phi^k\rangle_F\otimes\frac{1}{\sqrt{N}}\left(\sum_{i\neq k}|i\rangle_I - |k\rangle_I\right)
(UdisI)UfΦkF,I=ΦkF1N(ikiIkI)(U_{dis}\otimes\mathbb{I})U_f|\Phi^k\rangle_{F,I}=|\Phi^k\rangle_F\otimes\frac{1}{\sqrt{N}}\left(\sum_{i\neq k}|i\rangle_I - |k\rangle_I\right)

Variational Quantum Perceptron

Variational Quantum Perceptron

Quantum Ensemble Learning

Combine weak VQP \(\{V_t\}\) into strong classifier \(\rm{sign}(\sum_t |y_t - C_T|)\).

Required significant smaller size of training sets

Quantum Ensemble Learning

Query Complexity \(O(T\text{poly}(\log M\sqrt{\log N})\).

Runtime \(O(\text{poly}(\log M \log( M\sqrt{\log N}))\sqrt{\log N})\).

If \(|\mathcal{D}_t|\approx \log \sqrt{N} \)

Quantum Ensemble Learning

\( \mathcal{D}=\{\bm{x}_i,y_i\}_{i=1}^{10000} \)

\( \mathcal{D}_i=\{\bm{x}_k,y_k\}_{k=1}^{8} \)

Quantum Ensemble Learning

\( \mathcal{D}=\{\bm{x}_i,y_i\}_{i=1}^{10000} \)

\( \mathcal{D}_i=\{\bm{x}_k,y_k\}_{k=1}^{8} \)

Encoding requires 29 single and two qubits gates.

Oracles require 36 parameterized single qubit gates, 6 CNOT gates, 3 CZ gates, 24 Hadamard gates, and 3 Toffoli gates.

Bonus

The Expressive Power of Parameterized Quantum Circuits

What is the generative power of parametrized quantum circuits?

Thank you for your attention!