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→Y
f: X\to Y
Unknown Function
{(xi,yi)}i=1N
\{(x_i,y_i)\}_{i=1}^N
Training Data
H
\mathcal{H}
Hypothesis Set
Learning
Algorithm
f^
\hat{f}
Comp. Complexity
Sample Complexity
Quantum Computation
Classical Bit x∈Z={0,1}
QuBit ρ∈C2×2≥0 & Tr[ρ]=1
Random Bit (p(0)00p(1)) is a special case.
Quantum Computation
Quantum Operation: ρ↦σ
Unitary is a special case.
Quantum Measurement: ρ↦R
Quantum Challenge #1
Noncommutative: AB=BA
Moment Generating Function: Eeθ(A+B)=EeθAeθB
ba↦AB−1?
\frac{a}{b} \mapsto A B^{-1}?
ea+b↦eAeB?
e^{a+b} \mapsto e^A e^B?
Quantum Challenge #2
Entanglement: ρAB=ρA⊗ρB
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
In general, requires O(ϵ2rd) copies of ρ.
\text{In general, requires } O(\frac{rd}{\epsilon^2}) \text{ copies of } \rho.
Our readout improvement
Given: Input A∈Rm×n of rank r &
∣v⟩∈row(A)
Thm: poly(r,ϵ−1) query to QRAM &
poly(r,ϵ−1) copies of ∣v⟩.
[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⟩=∑i=1rxi∣Ag(i)⟩∈row(A)
2. quantum Gram-Schmidt Process algorithm to construct {Ag(i)}
3. Obtain {xi}.
Neural Networks
Expressive Power
⟩
⟩
⟩
[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).
Eθ∥∇θfTT∥≥O(n2−2L)
\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)
L(σG,D)=P(True∣σG)P(G)+P(False∣ρ)P(R),
\mathcal{L}(\sigma_G,\mathcal{D}) = P(\text{True}|\sigma_G)P(G) + P(\text{False}|\rho)P(R),
minσGmaxDL(σG,D)
\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).
Hydrogen Simulation
Thank you for your attention!
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