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

俞韋亘

賴青瑞

林俊吉

黃皓瑋

Unknown Function

Training Data

Hypothesis Set

Learning

Algorithm

Comp. Complexity

Sample Complexity

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: $\rho\mapsto\mathbb{R}$ 

Noncommutative: $AB\neq BA$ 

Moment Generating Function: $\mathbb{E}e^{\theta (A+B)}\neq\mathbb{E}e^{\theta A}e^{\theta B}$ 

Entanglement: $\rho_{AB}\neq \rho_{A}\otimes\rho_B$ 

CQ

QQ

QC

Readin

Readout

Q.C.

Expressivity

Trainability

Generalization

Learning

Model

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

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)

"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.

[1] Kaining Zhang, Min-Hsiu Hsieh, Liu Liu, Dacheng Tao. Toward Trainability of Quantum Neural Networks. arXiv:2011.06258 (2020).

Thm:

[1] ​Yuxuan Du, Min-Hsiu Hsieh, Tongliang Liu, Shan You, Dacheng Tao. On the learnability of quantum neural networks. arXiv:2007.12369 (2020)

$d$＝ $|\bm{\theta}|$

$T$＝ # of iteration

$L_Q$＝ circuit depth

$p$＝ error rate

$K$＝ # of measurements

$d$＝ $|\bm{\theta}|$

$T$＝ # of iteration

$L_Q$＝ circuit depth

$p$＝ error rate

$K$＝ # of measurements