Min-Hsiu Hsieh
Hon Hai (Foxconn) Quantum Computing Research Center
Challenge and Opportunity in Quantum Machine Learning
PME Special Quantum Seminar
Quantum Machine Learning
+
Why Quantum Computing?
Approximating the Jones polynomial is "BQP-complete".
Vaughan Jones - 1990 Fields Medal
- Aharonov, Jones, Landau, STOC 2006.
Why Machine Learning?
Unknown Function
Training Data
Hypothesis Set
Learning
Algorithm
Comp. Complexity
Sample Complexity
Unknown Function
Training Data
Hypothesis Set
Learning
Algorithm
Comp. Complexity
Sample Complexity
Quantum Ingredients
Quantum Advantage
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
Could QML achieve better end-to-end runtime?
QML Process
1. Readin
2. Readout
Many Challenges!
3.
Learning Machines
4. Noise
QRAM
- V. Giovannetti, S. Lloyd, L. Maccone, Phys. Rev. Lett. 100, 160501 (2008).
1.
Readin
Input Oracles for distribution
[1] Aleksandrs Belovs, Quantum Algorithms for Classical Probability Distributions, 27th annual European symposium on algorithms (esa 2019), 2019, pp. 16:1–16:11.
1.
Readin
There is no general readin protocol (with runtime guarantee) for arbitrary datasets.
1.
Readin
2.
Readout
State tomography:
Observation:
For ML problems, input and output have certain relationships.
2.
Readout
[1] Efficient State Read-out for Quantum Machine Learning Algorithms. Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Physical Review Research 3, 04395 (2021). [arXiv:2004.06421]
2.
Readout
poly(\(r,\epsilon^{-1}\)) query to QRAM.
Theorem:
Given:
\(-\) Input \(A\in\mathbb{R}^{m\times n}\) of rank \(r\)
\(-\) Output \( \bm{v} \in\text{row}(A)\)
\(-\) access to QRAM
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 \(\{x_i\}\).
2.
Readout
[1] Efficient State Read-out for Quantum Machine Learning Algorithms. Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Physical Review Research 3, 04395 (2021). [arXiv:2004.06421]
3.
Learning
Machine
Expressivity
Trainability
Generalization
Learning
Model
"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
3.1
Expressivity
\(\geq\)
[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].
3.1
Expressivity
\(\leq\)
\(\geq\)
"How easy is it to find the appropriate weights of the neural networks that fit the given data?"
3.2
Trainability
3.2
Trainability
Barren Plateau problem:
[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.
3.2
Trainability
Known BP Results
3.2
Trainability
Bad News for QML
-
Flat loss landscape.
-
Extremely small toleration to noise.
3.2
Trainability
Contribution 1:
BP free architecture
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Toward Trainability of Deep Quantum Neural Networks. [arXiv:2112.15002]
3.2
Trainability
Binary classification on the wine dataset (N=13)
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Toward Trainability of Deep Quantum Neural Networks. [arXiv:2112.15002]
3.2
Trainability
Contribution 2:
Initialization Matters
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Submitted to NIPS 2022.
3.2
Trainability
Finding the ground energy of the Ising model (N=15, L=10)
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Submitted to NIPS 2022.
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. PRX-Quantum 2, 040337 (2021)[arXiv:2007.12369]
3.2
Trainability
Contribution 3:
Trainability in ERM
\(d\)= \(|\bm{\theta}|\)
\(T\)= # of iteration
\(L_Q\)= circuit depth
\(p\)= error rate
\(K\)= # of measurements
3.2
Trainability
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. PRX-Quantum 2, 040337 (2021)[arXiv:2007.12369]
\(d\)= \(|\bm{\theta}|\)
\(T\)= # of iteration
\(L_Q\)= circuit depth
\(p\)= error rate
\(K\)= # of measurements
3.2
Trainability
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. PRX-Quantum 2, 040337 (2021)[arXiv:2007.12369]
3.3
Generalization
"Generalization refers to the model's ability to adapt properly to new, previously unseen data, drawn from the same distribution as the one used to train the model."
[1] S. Arunachalam, A. B. Grilo, and H. Yuen, arXiv:2002.08240 (2020).
3.3
Generalization
Separation between Learning models
3.3
Generalization
Contribution:
"Noisy QNN can efficiently simulate QSQ oracle."
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. PRX-Quantum 2, 040337 (2021)[arXiv:2007.12369]
經典: 1 error per 6 month in a 128MB PC100 SDRAM (2009)
量子: 1 error per second per qubit (2021)
4
Noise
\(\mathcal{C}\): The collection of all parameters
\(\mathcal{A}\): The collection of all possible circuits
\(\mathcal{E}_{\bm{a}}\): The error for the architecture \(\bm{a}\)
[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).
4.1
Error Mitigation
4.1
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).
Hydrogen Simulation+EM
4.1
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).
Could noise become useful in QML?
YES!
4.2
Harnessing Noise
4.2
Harnessing Noise
[Robustness]
[Privacy]
4.2.1
Providing Privacy
Differential Privacy (DP)
Classical DP is well studied; however, Quantum DP is not.
[1] Li Zhou and Mingsheng Ying. Differential privacy in quantum computation. In 2017 IEEE 30th Computer Security Foundations Symposium (CSF), pages 249–262. IEEE, 2017.
[2] Scott Aaronson and Guy N Rothblum. Gentle measurement of quantum states and differential privacy. Proceedings of ACM STOC‘2019.
4.2.1
Providing Privacy
Regression+DP
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. Quantum differentially private sparse regression learning. arXiv:2007.11921 (2020)
1. The first quantum DP algorithm.
2. Have the same privacy guarantee with the best classical DP algorithm.
3. Huge runtime improvement.
Contribution:
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. Quantum differentially private sparse regression learning. arXiv:2007.11921 (2020)
4.2.1
Providing Privacy
Adversarial Attack
[1] Lu et.al, “Quantum Adversarial Machine Learning". [arXiv:2001.00030]
4.2.2
Robustness
Adversarial Robustness
4.2.2
Robustness
2. Depolarizing noise suffices.
Contribution:
1. Explicit relation between p and \(\tau\).
[1]Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Dacheng Tao, Nana Liu. Quantum noise protects quantum classifiers against adversaries. Physical Review Research 3, 023153 (2021). [arXiv:2003.09416].
4.2.2
Robustness
Thank you for your attention!
Variational Quantum Circuits
By Lawrence Min-Hsiu Hsieh
Variational Quantum Circuits
AQIS 2021
- 96