Min-Hsiu Hsieh (謝明修)

鴻海研究院-量子計算研究所

Challenges and Opportunities of Quantum Machine Learning

"Quantum Computing" is a very young field.

Feynman, Richard (June 1982). "Simulating Physics with Computers"

"Let the computer itself be built of quantum mechanical elements which obey quantum mechanical laws"

Quantum Computers mostly likely will disprove Strong Church-Turing Thesis.

Why Quantum Computing?

New  Complexity Classes

Approximating the Jones polynomial is "BQP-complete".

Vaughan Jones - 1990 Fields Medal

- Aharonov, Jones, Landau, STOC 2006.

經典: 1 error per 6 month in a 128MB PC100 SDRAM  (2009)
量子: 1 error per second per qubit (2021)

Quantum Computer is not ready yet!

The Sampling task finished by '祖沖之' in about 1.2 HOURS will take the most powerful supercomputer at least 8 YEARS. [arXiv:2106.14734]

Quantum Supremacy

Math Is The Mother Of Science.

俞韋亘

賴青瑞

林俊吉

黃皓瑋

Why Machine Learning?

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 Computing 101

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 Operation: \(\rho\mapsto\sigma\) 

(Unitary is a special case.)

Quantum Measurement: \(\rho\mapsto\mathbb{R}\) 

Quantum Computing 101

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: \(\rho_{AB}\neq \rho_{A}\otimes\rho_B\) 

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

CQ
QC
Readin
Readout
Q.C.

Neural Networks

Expressivity
Trainability
Generalization

Learning

Model

Neural Network Expressivity

"how the architectural properties of a neural network (depth, width, layer type) affect the resulting functions it can compute"

[1] On the Expressive Power of Deep Neural  Networks. (ICML2017) arXiv:1606.05336

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].

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

"How easy is it to find the appropriate weights of the neural networks that fit the given data?"

Gradients vanish to zero exponentially with respect to the number of qubits.

[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.

f(\mathbf{\theta},\rho) =\frac{1}{2}\left(1+ \text{Tr}[O U(\theta)\rho U(\theta)^\dagger]\right)
\mathbb{E}_{\theta}\left(\frac{\partial f}{\partial \theta_j}\right)^2 =\epsilon \leq 2^{-\text{poly}(n)}

Barren Plateau problem

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}} \|^2\geq O(\frac{1+\log n}{n})

Thm:

Trainability of QNN in 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})
\mathcal{L}(\bm{\theta}):= \frac{1}{n}\sum_{j=1}^n \ell(y_i, \hat{y}_i) + r(\bm{\theta})
R_1\left(\bm{\theta}^{(T)}\right) := \mathbb{E} \left\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2
R_2\left(\bm{\theta}^{(T)}\right) := \mathbb{E}[\mathcal{L}(\bm{\theta}^{(T)})] - \mathcal{L}(\bm{\theta}^*)

Trainability of QNN in 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\|\nabla \mathcal{L}(\bm{\theta}^{(T)})\right\|^2
R_1 \leq \tilde{O}\left(poly\left(\frac{d}{T(1-p)^{L_Q}}, \frac{d}{BK(1-p)^{L_Q}} \right) \right)

\(d\)＝ \(|\bm{\theta}|\)

\(T\)＝ # of iteration

\(L_Q\)＝ circuit depth

\(p\)＝ error rate

\(K\)＝ # of measurements

Trainability of QNN in 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)

\(d\)＝ \(|\bm{\theta}|\)

\(T\)＝ # of iteration

\(L_Q\)＝ circuit depth

\(p\)＝ error rate

\(K\)＝ # of measurements

Thank you for your attention!

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