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 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?
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
CQ
QQ
CC
Sample Complexity for Learning Quantum Objects
Q. State
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).