Machine Learning Meets Quantum Computation
Min-Hsiu Hsieh (謝明修)
University of Technology Sydney
f:X→Y
f: X\to Y
Unknown Function
{(xi,yi)}i=1N
\{(x_i,y_i)\}_{i=1}^N
Training Data
H
\mathcal{H}
Hypothesis Set
Learning
Algorithm
f^
\hat{f}
Comp. Complexity
Sample Complexity
Quantum Computation
Classical Bit x∈Z={0,1}
QuBit ρ∈C2×2≥0 & Tr[ρ]=1
Random Bit (p(0)00p(1)) is a special case.
Quantum Computation
Quantum Operation: ρ↦σ
Unitary is a special case.
Quantum Measurement: ρ↦R
Quantum Challenge #1
Noncommutative: AB=BA
Moment Generating Function: Eeθ(A+B)=EeθAeθB
ba↦AB−1?
\frac{a}{b} \mapsto A B^{-1}?
ea+b↦eAeB?
e^{a+b} \mapsto e^A e^B?
Quantum Challenge #2
Entanglement: ρAB=ρA⊗ρB
Problem Setup
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
Alice
Bob
Compute (QS+RS+RT−QT)
Q
Q
R
R
S
S
T
T
Classical Mechanics
θ=(Q+R)S+(R−Q)T≤2
Let p(qrst):=Pr{Q=q,R=r,S=s,T=t}.
E[θ]=∑qrstp(qrst)(qs+rs+rt−qt)
\mathbb{E}[\theta]= \sum_{qrst}\text{p} (qrst)(qs+rs+rt-qt)
≤2
\leq 2
Probabilistically,
Quantum Mechanics
∣ΨAB⟩=21(∣0⟩A∣1⟩B−∣1⟩A∣0⟩B)
|\Psi_{AB}\rangle = \frac{1}{\sqrt{2}}\left(|0\rangle_A|1\rangle_B -|1\rangle_A |0\rangle_B\right)
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
={±1}
=\{\pm 1\}
Q
Q
R
R
S
S
T
T
Q=Z
Q=Z
R=X
R=X
S=2−Z−X
S=\frac{-Z-X}{\sqrt{2}}
T=2Z−X
T=\frac{Z-X}{\sqrt{2}}
Quantum Mechanics
E[θ]=⟨QS⟩+⟨RS⟩+⟨RT⟩−⟨QT⟩=22
\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
Ax=b
A=∑σℓ∣uℓ⟩⟨vℓ∣
x=∑λℓ∣vℓ⟩
Discrete Fourier Transform
yk=N1∑ℓ=0N−1eiN2πℓkxℓ
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 Θ(n2n) operations if N=2n.
Quantum Fourier Transform requires Θ(n2) operations if N=2n.
CQ
QQ
CC
Sample Complexity
{(xi,yi)}i=1N
\{(x_i,y_i)\}_{i=1}^N
Training Data
Rn(h)=N1∑i=1Nℓ(h(xi),yi)
R_n(h) = \frac{1}{N}\sum_{i=1}^N \ell (h(x_i), y_i)
h∈H
h\in\mathcal{H}
Hypothesis Set
f:X→Y
f: X\to Y
Unknown Function
Given a loss function
ℓ:Y×Y→R
\ell:Y\times Y \to \mathbb{R}
find
fn=argminh∈HRn(h)
{f}_n = \arg \min_{h\in \mathcal{H}} R_n (h)
where
Empirical Risk Minimization
if for any ϵ>0
Probably Approximately Correct (PAC) Learnable
H is PAC learnable
n→∞limμsupPr{h∈Hsup∣R(h)−Rn(h)∣>ϵ}=0
Sample Complexity
supμPr{suph∈HR(h)−Rn(h)≥ϵ}≤δ
Sample complexity mH(ϵ,δ) is the first quantity such that
for every n≥mH(ϵ,δ),
mH(ϵ,δ)=ϵ2C(VCdim(H)log(ϵ2)+log(δ2))
For Boolean functions 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=supf∈F∑i=1nf(xi)
Z=supf∈F∥∑i=1nf(Xi)∥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
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
fρ(E)=TrEρ
f_\rho(E) = \text{Tr} E\rho
Hypothesis Set
{fρ:ρ∈D(H)}
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
{(Ei,fρ(Ei)}i=1N
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
Hypothesis Set
{fρ:ρ∈D(H)}
\{f_{\rho}:\rho\in \mathcal{D}(\mathcal{H})\}
{(Ei,fρ(Ei)}i=1N
\{(E_i,f_\rho(E_i)\}_{i=1}^N
Training Data
Unknown Function
fρ:E(H)→R
f_\rho : \mathcal{E}(\mathcal{H}) \to \mathbb{R}
fatD(H)(ϵ,E(H))=O(logd/ϵ2)
Sample Complexity for Learning Quantum States
fE:D(H)→R
f_E: \mathcal{D}(\mathcal{H}) \to \mathbb{R}
fE(ρ)=TrEρ
f_E(\rho) = \text{Tr} E\rho
Hypothesis Set
{fE:E∈E(H)}
\{f_{E}:E\in \mathcal{E}(\mathcal{H})\}
{(ρi,fE(ρi)}i=1N
\{(\rho_i,f_E(\rho_i)\}_{i=1}^N
Training Data
Unknown Function
fE:D(H)→R
f_E : \mathcal{D}(\mathcal{H}) \to \mathbb{R}
Learning Unknown Measurement
Learning States
Learning Measurements
fatD(H)(ϵ,E(H))=O(logd/ϵ2)
fatE(H)(ϵ,D(H))=O(d/ϵ2)
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!
PhD Wanted!
Machine Learning Meets Quantum Computation Min-Hsiu Hsieh (謝明修) University of Technology Sydney