Challenges and Opportunities of Quantum Machine Learning
Min-Hsiu Hsieh (謝明修)
Hon Hai Quantum Computing Center
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
Problem Setup
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
Alice
Bob
Compute (QS+RS+RT−QT)
Q
Q
R
R
S
S
T
T
Classical Mechanics
θ=(Q+R)S+(R−Q)T≤2
Let p(qrst):=Pr{Q=q,R=r,S=s,T=t}.
E[θ]=∑qrstp(qrst)(qs+rs+rt−qt)
\mathbb{E}[\theta]= \sum_{qrst}\text{p} (qrst)(qs+rs+rt-qt)
≤2
\leq 2
Probabilistically,
Quantum Mechanics
∣ΨAB⟩=21(∣0⟩A∣1⟩B−∣1⟩A∣0⟩B)
|\Psi_{AB}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A|1\rangle_B -|1\rangle_A |0\rangle_B\right)
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
Q
Q
R
R
S
S
T
T
Q=Z
Q=Z
R=X
R=X
S=2−Z−X
S=\frac{-Z-X}{\sqrt{2}}
T=2Z−X
T=\frac{Z-X}{\sqrt{2}}
Quantum Mechanics
E[θ]=⟨QS⟩+⟨RS⟩+⟨RT⟩−⟨QT⟩=22
\mathbb{E}[\theta] = \langle QS\rangle + \langle RS\rangle + \langle RT\rangle - \langle QT\rangle= 2\sqrt{2}
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.
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).
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)
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)
θ∗=argminθ∈CL(θ,z)
\bm{\theta}^*= \arg \min_{\bm{\theta}\in\mathcal{C}} \mathcal{L}(\bm{\theta},\bm{z})
:=n1∑j=1nℓ(yi,y^i)+r(θ)
:= \frac{1}{n}\sum_{j=1}^n \ell(y_i, \hat{y}_i) + r(\bm{\theta})
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)
R1(θ(T)):=E[∇L(θ(T))2]
R_1\left(\bm{\theta}^{(T)}\right) := \mathbb{E}\left[\left\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2\right]
R1≤O~(poly(T(1−p)LQd,BK(1−p)LQd,(1−p)LQd))
R_1 \leq \tilde{O}\left(poly\left(\frac{d}{T(1-p)^{L_Q}}, \frac{d}{BK(1-p)^{L_Q}}, \frac{d}{(1-p)^{L_Q}} \right) \right)
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)
R2(θ(T)):=E[L(θ(T))]−L(θ∗)
R_2\left(\bm{\theta}^{(T)}\right) := \mathbb{E}[\mathcal{L}(\bm{\theta}^{(T)})] - \mathcal{L}(\bm{\theta}^*)
R2≤O~(poly(K2B(1−p)LQd+(1−p)LQd))
R_2\leq \tilde{O}\left( poly\left(\frac{d}{K^2B (1-p)^{L_Q}} + \frac{d}{(1-p)^{L_Q}}\right) \right)
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)
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
CQ
QQ
CC
Sample Complexity
{(xi,yi)}i=1N
\{(x_i,y_i)\}_{i=1}^N
Training Data
Rn(h)=N1∑i=1Nℓ(h(xi),yi)
R_n(h) = \frac{1}{N}\sum_{i=1}^N \ell (h(x_i), y_i)
h∈H
h\in\mathcal{H}
Hypothesis Set
f:X→Y
f: X\to Y
Unknown Function
Given a loss function
ℓ:Y×Y→R
\ell:Y\times Y \to \mathbb{R}
find
fn=argminh∈HRn(h)
{f}_n = \arg \min_{h\in \mathcal{H}} R_n (h)
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
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
fρ(E)=TrEρ
f_\rho(E) = \text{Tr} E\rho
Hypothesis Set
{fρ:ρ∈D(H)}
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
{(Ei,fρ(Ei)}i=1N
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
Hypothesis Set
{fρ:ρ∈D(H)}
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
{(Ei,fρ(Ei)}i=1N
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
fatD(H)(ϵ,E(H))=O(logd/ϵ2)
Sample Complexity for Learning Quantum States
fE:D(H)→R
f_E: \mathcal{D}(\mathcal{H}) \to \mathbb{R}
fE(ρ)=TrEρ
f_E(\rho) = \text{Tr} E\rho
Hypothesis Set
{fE:E∈E(H)}
\{f_{E}:E\in \mathcal{E}(\mathcal{H})\}
{(ρi,fE(ρi)}i=1N
\{(\rho_i,f_E(\rho_i)\}_{i=1}^N
Training Data
Unknown Function
fE:D(H)→R
f_E : \mathcal{D}(\mathcal{H}) \to \mathbb{R}
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).
Thank you for your attention!
Challenges and Opportunities of Quantum Machine Learning Min-Hsiu Hsieh (謝明修) Hon Hai Quantum Computing Center
Challenges and Opportunities of Quantum Machine Learning
By Lawrence Min-Hsiu Hsieh
Challenges and Opportunities of Quantum Machine Learning
CSIE, NTU. 16 July 2019
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