Min-Hsiu Hsieh
Hon Hai (Foxconn) Quantum Computing Research Center
Quantum Machine Learning
Quantum Seminar, Lisbon University
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
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
Quantum Ingredients
Quantum Advantage
Type of Input
Type of Algorithms
CQ
CC
QC
QQ
CQ
QQ
QC
-
Linear Equation Solvers
-
Peceptron
-
Recommendation Systems
-
Many Others (such as non-Convex Optimization)
-
State Tomography
-
Entanglement Structure
-
Quantum Control
How could QML achieve better end-to-end runtime?
QML Process
1. Readin
2. Readout
Many Challenges!
3.
Learning Machines
4. Noise
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.
O_p |0\rangle =\sum_{x\in\mathcal{X}} \sqrt{p_x} |x\rangle_A\otimes|\phi_x\rangle_B
O_p |0\rangle =\sum_{x\in\mathcal{X}} \sqrt{p_x} |x\rangle
O_{s} |0\rangle =\sum_{x\in\mathcal{X}} n^{-1/2} |x\rangle \otimes |\#_s(x)\rangle
1.
Readin
QRAM
- V. Giovannetti, S. Lloyd, L. Maccone, Phys. Rev. Lett. 100, 160501 (2008).
1.
Readin
There is no general readin protocol (with nontrivial space-time guarantee) for arbitrary datasets.
1.
Readin
Ongoing Progress:
1.
Readin
Could we find an approximate amplitude encoding (AAE) for real-world datasets with polynomial gates?
Entropy reduction
Ongoing Progress:
1.
Readin
Text
2.
Readout
\text{require } O(\frac{rd}{\epsilon^2}) \text{ copies of } \rho.
State tomography:
- \ r \text{ is the rank of } \rho.
- \ d \text{ is the dimension.}
Observation:
For ML problems, input and output have certain relationships.
- A \bm{x} =\bm{b}.
- A = \sum_{i} \sigma_i \bm{u}_i \bm{v}^\dagger_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]
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:
\mathbb{E}_{\theta}\|\triangledown_{\mathbf{\theta}} f \|^ 2 =\epsilon \leq 2^{-\text{poly}(n)}
\text{where } f(\mathbf{\theta},\rho) =\text{Tr}[O U(\theta)\rho U(\theta)^\dagger].
[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
\mathbb{E}_{\theta}\|\triangledown_{\mathbf{\theta}} f \|^ 2 \geq c(\rho_{in}) 2^{-3LS}
\text{where } {\bf{\theta}} \sim [0,2\pi].
[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
\mathbb{E}_{\bf{\theta}}\|\triangledown_{\bf{\theta}} f \|^ 2 \geq c(\rho_{in}) \text{poly}(L)^{-1}
\text{where } {\bf{\theta}} \sim N(0, \frac{1}{4S(L+2)}).
[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. NeurIPS 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. 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]
\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}^*)
3.2
Trainability
Contribution 3:
Trainability in ERM
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
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]
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
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]
3.4
Quantum
Advantage
Any quantum advantage could be achieved using QML?
->No universal quantum advantage.
Qian, Y., Wang, X., Du, Y., Wu, X., & Tao, D. (2022). The dilemma of quantum neural networks. IEEE Transactions on Neural Networks and Learning Systems.
3.4
Quantum
Advantage
How to guarantee problem-dependent quantum advantage using QNN?
->We identify conditions when the expected risk in classification tasks is minimized.
Yuxuan Du, Yibo Yang, Dacheng Tao, Min-Hsiu Hsieh. "Demystify Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification." Physical Review Letters 131, 140601 (2023)
3.4
Quantum
Advantage
Contribution:
Yuxuan Du, Yibo Yang, Dacheng Tao, Min-Hsiu Hsieh. "Demystify Problem-Dependent Power of Quantum Neural Networks on Multi-Class Classification." Physical Review Letters 131, 140601 (2023)
經典: 1 error per 6 month in a 128MB PC100 SDRAM (2009)
量子: 1 error per second per qubit (2021)
4
Noise
(\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 \(\bm{a}\)
[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. npj Quantum Information volume 8, Article number: 62 (2022).
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. npj Quantum Information volume 8, Article number: 62 (2022).
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