Challenges and Opportunities of Quantum Machine Learning
Min-Hsiu Hsieh (謝明修)
Hon Hai Quantum Computing Center
Unknown Function
Training Data
Hypothesis Set
Learning
Algorithm
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
Quantum Challenge #2
Entanglement: ρAB=ρA⊗ρB
Problem Setup
Alice
Bob
Compute (QS+RS+RT−QT)
Classical Mechanics
θ=(Q+R)S+(R−Q)T≤2
Let p(qrst):=Pr{Q=q,R=r,S=s,T=t}.
Probabilistically,
Quantum Mechanics
Quantum Mechanics
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
Readout
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).
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)
Trainability of QNN: ERM
[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: ERM
[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: ERM
[1] Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)
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.
QQ
-
Entanglement Test
with Jian-Wei Pan's group (submitted)
Quantum Generative and Adversarial Networks (QGAN)
[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
CQ
QQ
CC
Sample Complexity
Training Data
Hypothesis Set
Unknown Function
Given a loss function
find
where
Empirical Risk Minimization
if for any ϵ>0
Probably Approximately Correct (PAC) Learnable
H is PAC learnable
n→∞limμsupPr{h∈Hsup∣R(h)−Rn(h)∣>ϵ}=0
Sample Complexity
supμPr{suph∈HR(h)−Rn(h)≥ϵ}≤δ
Sample complexity mH(ϵ,δ) is the first quantity such that
for every n≥mH(ϵ,δ),
mH(ϵ,δ)=ϵ2C(VCdim(H)log(ϵ2)+log(δ2))
For Boolean functions H
[1] Vapnik, Springer-Verlag, New York/Berlin, 1982.
[2] Blumer, Ehrenfeucht, Haussler, and Warmuth, Assoc. Comput. Machine, vol. 36, no. 4, pp. 151--160, 1989.
Z=supf∈F∑i=1nf(xi)
Z=supf∈F∥∑i=1nf(Xi)∥p.
There are only very limited matrix concentration results!!
[1] Joel Tropp. User-friendly tail bounds for sums of random matrices. arXiv:1004.4389.
Sample Complexity for Learning Quantum Objects
Q. State
Measurement
Hypothesis Set
Training Data
Unknown Function
Hypothesis Set
Training Data
Unknown Function
fatD(H)(ϵ,E(H))=O(logd/ϵ2)
Sample Complexity for Learning Quantum States
Hypothesis Set
Training Data
Unknown Function
Learning Unknown Measurement
Learning States
Learning Measurements
fatD(H)(ϵ,E(H))=O(logd/ϵ2)
fatE(H)(ϵ,D(H))=O(d/ϵ2)
Hao-Chung Cheng, MH, Ping-Cheng Yeh. The learnability of unknown quantum measurements. QIC 16(7&8):615–656 (2016).