Challenges and Opportunities of Quantum Machine Learning
Min-Hsiu Hsieh (謝明修)
Hon Hai Quantum Computing Center
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: \(\rho_{AB}\neq \rho_{A}\otimes\rho_B\)
Problem Setup
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
Alice
Bob
Compute \((QS+RS+RT-QT)\)
Q
R
S
T
Classical Mechanics
\(\theta=(Q+R)S+(R-Q)T\leq 2\)
Let \(\text{p}(qrst) := \text{Pr}\{Q=q,R=r,S=s,T=t\}\).
\mathbb{E}[\theta]= \sum_{qrst}\text{p} (qrst)(qs+rs+rt-qt)
\leq 2
Probabilistically,
Quantum Mechanics
|\Psi_{AB}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A|1\rangle_B -|1\rangle_A |0\rangle_B\right)
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
=\{\pm 1\}
Q
R
S
T
Q=Z
R=X
S=\frac{-Z-X}{\sqrt{2}}
T=\frac{Z-X}{\sqrt{2}}
Quantum Mechanics
\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
\text{In general, requires } O(\frac{rd}{\epsilon^2}) \text{ copies of } \rho.
Readout
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)
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)
\bm{\theta}^*= \arg \min_{\bm{\theta}\in\mathcal{C}} \mathcal{L}(\bm{\theta},\bm{z})
:= \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)
R_1\left(\bm{\theta}^{(T)}\right) := \mathbb{E}\left[\left\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2\right]
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)
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)
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)
\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).
Hydrogen Simulation
CQ
QQ
CC
Sample Complexity
\{(x_i,y_i)\}_{i=1}^N
Training Data
R_n(h) = \frac{1}{N}\sum_{i=1}^N \ell (h(x_i), y_i)
h\in\mathcal{H}
Hypothesis Set
f: X\to Y
Unknown Function
Given a loss function
\ell:Y\times Y \to \mathbb{R}
find
{f}_n = \arg \min_{h\in \mathcal{H}} R_n (h)
where
Empirical Risk Minimization
if for any \(\epsilon>0\)
Probably Approximately Correct (PAC) Learnable
\(\mathcal{H}\) is PAC learnable
$$ \lim_{n\to\infty}\sup_{\mu} \Pr\{\sup_{h\in\mathcal{H}}|R(h) - R_n(h)| >\epsilon\} = 0$$
Sample Complexity
\(\sup_{\mu} \Pr \left\{ \sup_{h\in\mathcal{H}} \big|R(h)-R_n(h)\big|\geq \epsilon \right\}\leq \delta\)
Sample complexity \(m_\mathcal{H}(\epsilon,\delta)\) is the first quantity such that
for every \(n\geq m_\mathcal{H}(\epsilon,\delta),\)
\(m_{\mathcal{H}}(\epsilon,\delta)= \frac{C}{\epsilon^2}\left(\text{VCdim}(\mathcal{H})\log\left(\frac{2}{\epsilon}\right)+\log\left(\frac{2}{\delta}\right)\right)\)
For Boolean functions \(\mathcal{H}\)
[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.
\(Z=\sup_{f\in\mathcal{F}}\big| \sum_{i=1}^n f(x_i)\big|\)
\({Z}=\sup_{\bm{f}\in\mathcal{F}}\left\| \sum_{i=1}^n \bm{f}(\bm{X}_i)\right\|_p.\)
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
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
f_\rho(E) = \text{Tr} E\rho
Hypothesis Set
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
Hypothesis Set
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
fat\(_{\mathcal{D}(\mathcal{H})}(\epsilon,\mathcal{E}(\mathcal{H})) = O(\log d/\epsilon^2)\)
Sample Complexity for Learning Quantum States
f_E: \mathcal{D}(\mathcal{H}) \to \mathbb{R}
f_E(\rho) = \text{Tr} E\rho
Hypothesis Set
\{f_{E}:E\in \mathcal{E}(\mathcal{H})\}
\{(\rho_i,f_E(\rho_i)\}_{i=1}^N
Training Data
Unknown Function
f_E : \mathcal{D}(\mathcal{H}) \to \mathbb{R}
Learning Unknown Measurement
Learning States
Learning Measurements
fat\(_{\mathcal{D}(\mathcal{H})}(\epsilon,\mathcal{E}(\mathcal{H})) = O(\log d/\epsilon^2)\)
fat\(_{\mathcal{E}(\mathcal{H})}(\epsilon,\mathcal{D}(\mathcal{H})) = O( d/\epsilon^2)\)
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
By Lawrence Min-Hsiu Hsieh
Challenges and Opportunities of Quantum Machine Learning
CSIE, NTU. 16 July 2019
