The Power of Quantum Neural Networks
Min-Hsiu Hsieh
Hon Hai Quantum Computing Center, Taiwan
Why Quantum Computing ?
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
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
CQ
QC
Readin
Readout
Q.C.
Readout
\text{In general, requires } O(\frac{rd}{\epsilon^2}) \text{ copies of } \rho.
Our readout improvement
State Tomography:
Given: Input \(A\in\mathbb{R}^{m\times n}\) of rank \(r\) &
\(|v\rangle \in\text{row}(A)\)
Thm:
poly(\(r,\epsilon^{-1}\)) copies of \(|v\rangle\).
[1] Efficient State Read-out for Quantum Machine Learning Algorithms. Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. arXiv:2004.06421
High Level Proof
1. \(|v\rangle = \sum_{i=1}^r x_i |A_{g(i)}\rangle\in\text{row}(A)\)
2. quantum Gram-Schmidt Process algorithm to construct \(\{A_{g(i)}\}\)
3. Obtain \(\{x_i\}\).
Neural Networks
Expressivity
Trainability
Generalization
Learning
Model
Neural Network Expressivity
"how the architectural properties of a neural network (depth, width, layer type) affect the resulting functions it can compute"
[1] On the Expressive Power of Deep Neural Networks. (ICML2017) arXiv:1606.05336
Expressive Power
\(\rangle\)
\(\rangle\)
\(\rangle\)
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao. The Expressive Power of Parameterized Quantum Circuits. Physical Review Research 2, 033125 (2020) [arXiv:1810.11922].
Learnability of QNN
Learnability = trainability + generalization
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
Trainability of QNN
"How easy is it to find the appropriate weights of the neural networks that fit the given data?"
Gradients vanish to zero exponentially with respect to the number of qubits.
[1] Jarrod R McClean, Sergio Boixo, Vadim N Smelyanskiy, Ryan Babbush, and Hartmut Neven. Barren plateaus in quantum neural network training landscapes. Nature communications, 9(1):1– 6, 2018.
f(\mathbf{\theta},\rho) =\frac{1}{2}\left(1+ \text{Tr}[O U(\theta)\rho U(\theta)^\dagger]\right)
\mathbb{E}_{\theta}\left(\frac{\partial f}{\partial \theta_j}\right)^2 =\epsilon \leq 2^{-\text{poly}(n)}
Barren Plateau problem
Trainability of QNN
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Toward Trainability of Quantum Neural Networks. arXiv:2011.06258 (2020).
\mathbb{E}_{\bm{\theta}} \|\nabla_{\bm{\theta}} f_{\text{TT}} \|^2\geq O(\frac{1+\log n}{n})
Thm:
Trainability of QNN in ERM
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
\bm{\theta}^*= \arg \min_{\bm{\theta}\in\mathcal{C}} \mathcal{L}(\bm{\theta},\bm{z})
\mathcal{L}(\bm{\theta}):= \frac{1}{n}\sum_{j=1}^n \ell(y_i, \hat{y}_i) + r(\bm{\theta})
R_1\left(\bm{\theta}^{(T)}\right) := \mathbb{E} \left\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2
R_2\left(\bm{\theta}^{(T)}\right) := \mathbb{E}[\mathcal{L}(\bm{\theta}^{(T)})] - \mathcal{L}(\bm{\theta}^*)
Trainability of QNN in ERM
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
R_1\left(\bm{\theta}^{(T)}\right) := \mathbb{E} \left\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2
R_1 \leq \tilde{O}\left(poly\left(\frac{d}{T(1-p)^{L_Q}}, \frac{d}{BK(1-p)^{L_Q}} \right) \right)
\(d\)＝ \(|\bm{\theta}|\)
\(T\)＝ # of iteration
\(L_Q\)＝ circuit depth
\(p\)＝ error rate
\(K\)＝ # of measurements
Trainability of QNN in ERM
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
R_2\left(\bm{\theta}^{(T)}\right) := \mathbb{E}[\mathcal{L}(\bm{\theta}^{(T)})] - \mathcal{L}(\bm{\theta}^*)
R_2\leq \tilde{O}\left( poly\left(\frac{d}{K^2B (1-p)^{L_Q}} ,\frac{d}{(1-p)^{L_Q}}\right) \right)
\(d\)＝ \(|\bm{\theta}|\)
\(T\)＝ # of iteration
\(L_Q\)＝ circuit depth
\(p\)＝ error rate
\(K\)＝ # of measurements
Generalization of QNN
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
Thm:
Quantum Statistical Query algorithms can be efficiently simulated by QNN.
Two Applications of QNN
QQ
-
Entanglement Test
with Jian-Wei Pan's group (submitted)
Quantum Generative and Adversarial Networks (QGAN)
\mathcal{L}(\sigma_G,\mathcal{D}) = P(\text{True}|\sigma_G)P(G) + P(\text{False}|\rho)P(R),
\min_{\sigma_G}\max_{\mathcal{D}}\mathcal{L}(\sigma_G,\mathcal{D})
[1] Lloyd, S. & Weedbrook, C. Quantum generative adversarial learning. Physical review letters 121, 040502 (2018).
Results
Error Mitigation
[1] Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, Dacheng Tao. Quantum circuit architecture search: error mitigation and trainability enhancement for variational quantum solvers. arXiv:2010.10217 (2020).
(\bm{\theta}^*,\bm{a}^*)= \arg \min_{\bm{\theta}\in\mathcal{C},\bm{a}\in\mathcal{A}} \mathcal{L}(\bm{\theta},\bm{a}, \mathcal{E}_{\bm{a}})
\(\mathcal{C}\): The collection of all parameters
\(\mathcal{A}\): The collection of all possible circuits
\(\mathcal{E}_{\bm{a}}\): The error for the architecture \(\bm{a}\)
Error Mitigation
Yuxuan Du, Tao Huang, Shan You, Min-Hsiu Hsieh, Dacheng Tao. Quantum circuit architecture search: error mitigation and trainability enhancement for variational quantum solvers. arXiv:2010.10217 (2020).
Hydrogen Simulation
Gradient
CQ
QQ
CC
Thank you for your attention!
The power of quantum neural networks
By Lawrence Min-Hsiu Hsieh
The power of quantum neural networks
MLQ 2021
