Machine Learning Meets Quantum Computation
Min-Hsiu Hsieh (謝明修)
University of Technology Sydney
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
\(A\mathbf{x} = \mathbf{b}\)
\(A =\sum \sigma_\ell |u_\ell\rangle\langle v_\ell|\)
\(\mathbf{x} =\sum \lambda_\ell |v_\ell\rangle\)
Discrete Fourier Transform
y_k = \frac{1}{\sqrt{N}}\sum_{\ell=0}^{N-1} e^{i\frac{2\pi}{N} \ell k} x_\ell
Classical Fast Fourier Transform requires \(\Theta (n2^n)\) operations if \(N=2^n\).
Quantum Fourier Transform requires \(\Theta (n^2)\) operations if \(N=2^n\).
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).