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|
P_s= |+\rangle\langle +|^{\otimes n}
U_s= 2|+\rangle\langle +|^{\otimes n} - I
U_\omega= 2|\omega\rangle\langle \omega| - I
How to identify
from
Grover Search
Recommendation Systems
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
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}
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
|\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).
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)
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.
Part II
The Expressive Power of Parameterized Quantum Circuits
What is the generative power of parametrized quantum circuits?
MPQC
- arXiv:1801.07686, arXiv:1804.04168
TPQC
arXiv:1803.11537
Boltzmann Machine
Boltzmann Machine
Part III
Quantum Divide-and-Conquer GAN
Seth Lloyd and Christian Weedbrook. "Quantum generative adversarial learning". Phys. Rev. Lett. 121, 040502 (2018)
Generative Adversarial Network
Thank you for your attention!
Quantum Machine Learning
By Lawrence Min-Hsiu Hsieh
