Challenges and Opportunities of Quantum Machine Learning
謝明修
雪梨科技大學->鴻海研究院
第一屆台灣量子科技共識論壇
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 Computation
Classical Bit \(x\in\mathbb{Z}=\{0,1\}\)
QuBit \(\rho\in\mathbb{C}^{2\times 2}\geq0\) & Tr\([\rho]=1\)
Random Bit \(\left(\begin{array}{cc} p(0) & 0\\ 0 & p(1) \end{array}\right)\) is a special case.
Quantum Computation
Quantum Operation: \(\rho\mapsto\sigma\)
Unitary is a special case.
Quantum Measurement: \(\rho\mapsto\mathbb{R}\)
Quantum Challenge #1
Noncommutative: \(AB\neq BA\)
Moment Generating Function: \(\mathbb{E}e^{\theta (A+B)}\neq\mathbb{E}e^{\theta A}e^{\theta B}\)
\frac{a}{b} \mapsto A B^{-1}?
e^{a+b} \mapsto e^A e^B?
Quantum Challenge #2
Entanglement: \(\rho_{AB}\neq \rho_{A}\otimes\rho_B\)
Why Quantum Computation Matters?
Many More!
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
CC
-
Linear Equation Solvers
-
Recommendation Systems
-
Semidefinite Programming
-
Minimum Conical Hull
Quantum-Inspired Classical Algorithms
CQ
QC
Readin
Readout
Q.C.
Input Models
[1] V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 100, 160501 (2008).
Readout
\text{In general, requires } O(\frac{rd}{\epsilon^2}) \text{ copies of } \rho.
Our readout improvement
Given: Input \(A\in\mathbb{R}^{m\times n}\) of rank \(r\) &
\(|v\rangle \in\text{row}(A)\)
Thm: poly(\(r,\epsilon^{-1}\)) query to QRAM &
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
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].
Trainability of QNN
Gradients vanish to zero exponentially with respect to the number of qubits.
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.
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}} \|\geq O(\frac{2^{-2L}}{n})
Thm:
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)
QQ
-
Entanglement Test
with Jian-Wei Pan's group (in preparation)
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
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).
