introduction to quantum machine learning m. hilke (quantum … · 2017. 11. 14. · =10b tb worth...
TRANSCRIPT
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Introduction to Quantum Machine LearningM. Hilke
(Quantum Nano Electronics Laboratory)
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Why Quantum Machine Learning?
Hype Curve
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Why Quantum Machine Learning?
Money?
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Structure:
• Machine Learning• Quantum Machine Learning
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Machine Learning:
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Matlab demo
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Idea of Machine Learning:
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me
Neuron
Axon
20mnot me
Neuron network:1011 neurons and 1014 synapses in the human brain3x1011 neurons in an elephant brain
(six-core i7 has 109 transistors)
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Principle of Deep Convolution Neural Network for Face Recognition
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Input layer
Input picture into neural network
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Input layer layer 1
Add neuron layer
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Input layer layer 1
layer 2 Add more layers
Deep Neural Network (more than 1 layer)
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Input layer layer 1
layer 2
Output layer
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Input layer layer 1
layer 2
Output layer
Artificial Neuron
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layer 1 layer 2
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
Value of neurons at layer 1
General Computation Flow
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
𝑊12(5)
𝑊12(1)
Value of neurons at layer 1
𝑊12(1)
axon weight
axon weight between layer 1 (neuron 5) and layer 2
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
𝑥
𝑊12(5)
𝑊12(1)
Value of neurons at layer 1
𝑊12(1)
axon weight
𝑥 =
𝑛=1
5
𝑊12(𝑛)𝐿1(𝑛)
neuron input (layer 2)
axon weight between layer 1 (neuron 5) and layer 2
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
𝑥
𝑊12(5)
𝑊12(1)
Value of neurons at layer 1
𝑊12(1)
axon weight
𝑥 =
𝑛=1
5
𝑊12(𝑛)𝐿1(𝑛)
neuron input (layer 2)
b2
Threshold value of neuron
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
b2
Threshold value of neuron
𝑥 =
𝑛=1
5
𝑊12(𝑛)𝐿1(𝑛)
neuron output
𝑥
𝐿2 =1
1 + 𝑒−𝑥+𝑏2
𝐿2
neuron input𝐿2
𝑥𝑏2
𝑊12(5)
𝑊12(1)
axon weight
(0 < 𝐿2 < 1)
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layer 1 layer 2
𝐿1(1)
𝐿1(2)
𝐿1(3)
𝐿1(4)
𝐿1(5)
𝑥 =
𝑛=1
5
𝑊12(𝑛)𝐿1(𝑛)
neuron output
𝑥
𝐿2 =1
1 + 𝑒−𝑥+𝑏2
𝐿2
neuron input𝐿2
𝑥𝑏2𝑊12
(1)
axon weight
(0 < 𝐿2 < 1)
Convolution layer
Fully Connected Layers
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I1
I2
I3
I4
Input layer layer 1
layer 2
Ԧ𝐼
𝐿2 =1
1 + exp(−𝑊12𝐿1 + 𝑏2)
𝐿1
𝐿2
𝐿1, 𝑏1
𝑊12
𝐿2, 𝑏2
𝑊𝐼1
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I1
I2
I3
I4
Input layer layer 1
layer 2
Output layer
Ԧ𝐼
𝐿2 =1
1 + exp(−𝑊12𝐿1 + 𝑏2)
𝐿1
𝐿2
𝑂
𝐿1, 𝑏1
𝑊12
𝐿2, 𝑏2
𝑊𝐼1 𝑊2𝑂
Largest Output wins!
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Learning Phase of Neural Network requires large amounts of training data and can take a lot of processing time.
Recognizing 1 picture: <0.1s (fast)
Typically: training > 1hr and evaluating < 0.1s for “simple” (few tasks) neural network (for a laptop).
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Monkey faces training data (~1,000 pictures)
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Female faces training data (~1,000 pictures)
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Female faces training data (~1,000 pictures)
Male faces training data (~1,000 pictures)
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Input layer layer 1
layer 2
Output layer
𝑊12(11)
𝑊2𝑂(11)
𝑊12(56)
𝑏1(5)
𝑏2(6)
𝑏1(4)
𝑏1(3)
𝑏1(2)
𝑏1(1)
𝑏2(5)
𝑏2(4)
𝑏2(3)
𝑏2(2)
𝑏2(1)
𝑊12(12)
𝑊12(12)
Training network means finding optimal b and W
𝑊𝐼1(11)
𝑏𝑂(3)
𝑏𝑂(2)
𝑏𝑂(1)
𝑏𝐼(4)
𝑏𝐼(3)
𝑏𝐼(2)
𝑏𝐼(1)
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Training of Deep Convolution Neural Network
Minimize the cost function: (quadratic)
desired output
NN output
(cross-entropy)
Update NN parameters:
C
iterations
learning rate
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Once Network is trained it is very powerful for specific tasks:
2014: deep face (Facebook AI Research) – close to human performance for face recognition
2016: AlphaGo was developed by the Google DeepMind team and beats humans in Go
But! Takes a lot of time to find the best 100 million parameters
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Good resources to do it yourself:
http://neuralnetworksanddeeplearning.com/
Good introduction with ML code in python
1)
2) http://www.deeplearningbook.org/
By the master (Yoshua Bengio, Goodfellow and Courville)
3) Open source Matlab CNN code
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What about ?
Srce: backreaction
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Classical image:
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Classical image:
16 x16 pixels with 256 grey tones = 65536 worth of data or 8kB uncompressed (bmp).(~1kB compressed)
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Quantum image:
16 x16 spins ½ (qubits)
Spin up Spin down
=256 qubits
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Spin up Spin down
=256 qubits
Quantum image:
16 x16 spins ½ (qubits) = 2(16x16) = 1077 worth of data.
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Spin up Spin down
The world now has 1023
=10B TB worth of digital data (internet, hard drives, DVDs,…)
=256 qubits
To describe 256 qubits classically one needs 1076 classical Bytes(1 Byte=8 bits)
Quantum image:
16 x16 spins ½ (qubits) = 2(16x16) = 1077 worth of data.
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Quantum images:(states)
Very hard to store or to compute with a classical computer
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Quantum Machine Learning
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Quantum Machine Learning
1) Quantum data – classical machine2) Classical data – quantum machine3) Quantum data – quantum machine
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Input layer layer 1
layer 2
Output layer
|𝜑 >
1) Quantum data – classical machine
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V+V-
V0
V0I
q=-e
Flow of electrons through a disordered conductor
Simple example of Quantum Classical Machine Learning
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V+V-
V0
V0I
q=-e
Flow of electrons through a disordered conductor
Microscope of electrons (scanning probe)
The Ginger Lab
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V0V0
V0Microscope of electrons (scanning probe)
The Ginger Lab
V0
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Disorder potential Electron density
Quantum Calculation (solving Schrödinger equation) and computing the Local Density of State at E0
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This is the electron density, what is the corresponding potential?
=> Hard problem
Can Quantum machine learning help?
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HardEasy
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Potential 4
Potential 3
Potential 2
Potential 1
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Matlab quantum machine learning (over 90% accuracy)
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LDOS for different disorder configurations for same disorder amplitude
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Input layer layer 1
layer 2
Output layer
|𝜑 >
1) Quantum data – classical machine
Classicize:𝜑 𝜑
V
WORKS!
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1) Quantum data – classical machine (other example)
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(Photoionization detector)
Source: Mike Williams
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Source: Mike Williams
Delta Log (likelihood)
Neural Network ML
(Photoionization detector)
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Input Output
2) classical data – quantum machine
(Similar goal to quantum computing: enhance efficiency by using a quantum computer)
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Quantum principal component analysis (an example)
Comparing Stocks
Yesterday’s data
𝑣𝑡𝑛 = stock n change at time = t
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jsandatascience.comCovariance of stock change:
CISCO Chevron Exon Mobile
Quantum principal component analysis (an example)
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𝑣𝑡𝑛 = change of stock n at time t
𝑣𝑡 = vector for N stocks → |𝑣𝑡 > : quantum state
Density matrix
QPCA: Find Eigenvalues in O(logN)2 instead of O(N2) for classical PCA
Use in Quantum Machine Learning Software for speed-up
Quantum principal component analysis (an example)
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Input
Output
3) quantum data – quantum machine
|𝜑 >
|𝜓 >
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(3.1) Superposition of memorized states (Quantum Associative Memory - Ventura and Martinez ‘98)
Idea: Create a superposed memory state of learned states
with
Requires copies of |M> since the state is destroyed after the probabilistic measurement
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(3.2) Time evolution (f.ex interacting quantum dots – Behrman and co-workers ’99 or Perus ‘00)
inputoutput
Green’s function of the trained system
Ex: interacting quantum dots
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(3.3) Time flow approaches (Kak ’95, Zak and Williams ’98, Gupta and Zia ‘01,…)
(a) Quantum Measurement: after some time a quantum measurement is performed – then time evolution –measurement -…
(b) Dissipative operator: after some time a dissipative operator is applied and successive time evolution and dissipative operator…
(c) Successive entanglement: Panella and Martinelli ‘11
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(3.4) Quantum Boltzmann Machine (quantization of the classical Boltzmann Machine)
Classical Restricted Boltzmann Machine:
v hw Probabilistic machine: probability value of every
state determined by the local energy 𝐸𝑖 = 𝑧𝑖 +
σ𝑗𝑊𝑖𝑗𝑧𝑗 ; 𝑧 = 𝑣 𝑜𝑟 ℎ ; 𝑃 𝑧𝑖 = 1 =1
1+𝑒−𝐸𝑖. This
will eventually minimize global energy
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(3.4) Quantum Boltzmann Machine (quantization of the classical Boltzmann Machine)
output
Learning of Restricted Boltzmann Machine:
1. Clamp input and desired output (visible layer) => find global minimum
2. Clamp only input => find global minimum => compare output with desired output. Adjust weights and biases by optimizing difference between output and desired output.
3. Use your machine
input
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(3.4) Quantum Boltzmann Machine (quantization of the classical Boltzmann Machine)
From Crawford et al. ‘16Deep machine
input
input
output
output
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(3.4) Quantum Boltzmann Machine (quantization of the classical Boltzmann Machine)
Deep machineinput
input
output
output
Quantization of Boltzmann Machine:
From Crawford et al. ‘16
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Jason Rolfe Roger MelkoBohdan KulchytskyyEvgeny Andriyash
arXiv:1601.02036
Slide from AminSlide from Amin
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Transverse Ising Hamiltonian
Slide from Amin
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Quantum Boltzmann Distribution
Boltzmann probability distribution:
Density matrix:
Projection operator Identity matrix
Slide from Amin
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Training
Clamped average Unclamped average
Slide from Amin
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Copyright© 2016, D-Wave Systems Inc.
Quantum Boltzmann Machine
Classical BM
Bound gradient
D=2
Exact gradient
(D is trained)
D final = 2.5
Train a Boltzmann machine using quantum Boltzmann
distribution (Amin, Andriyash, et al., arXiv:1601.02036)
Slide from Amin
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=𝜀𝑖 𝑡𝑖𝑡𝑖 −𝜀𝑖
(i)𝐻 =
𝑖=1
𝑁
𝕀⨂⋯⨂𝐻𝑖 ⨂⋯⨂𝕀 +
𝑖𝑗
𝑉𝑖𝑗
= 𝜀𝑖𝜎𝑧 + 𝑡𝑖𝜎
𝑥
= 𝐻𝑖
In general: some collection of interacting qubits
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In general: some collection of interacting qubits
For quantum machine learning need an input and output subset:
Input
Output
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In general: some collection of interacting qubits
For quantum machine learning need an input and output subset:
Input
Output
Connected to quantum states
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How can this be modeled?
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=𝜀𝑖 𝑡𝑖𝑡𝑖 −𝜀𝑖
(i)𝐻 =
𝑖=1
𝑁
𝕀⨂𝕀⨂𝕀⨂𝑆𝑖⨂𝕀⨂𝕀 +
𝑖𝑗
𝑉𝑖𝑗
= 𝜀𝑖𝜎𝑧 + 𝑡𝑖𝜎
𝑥
= 𝑆𝑖
⟺
𝐻 =
𝑖=1
2𝑁
𝜖𝑖|𝑖 >< 𝑖| +
𝑖𝑗
2𝑁
𝑡𝑖𝑗 |𝑖 >< 𝑗|
Collection of qubits
Highly connected tight binding model, which can be computed classically.
⋯𝑖⋯
(6 qubits)
(64 sites)
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Quantum Machine Learning
1) Quantum data – classical machineMany useful applications. Can use powerful classical ML codes (Deep Convolution NN). Often outperform non-ML approaches.
2) Classical data – quantum machineSome powerful algorithms exist but many questions remain, particularly for the learning phase.
1) Quantum data – quantum machineMany different preliminary approaches, but it’s just the beginning. No clear emerging winning candidate. There is a lot of fundamental work remaining to be done.
Thanks!