Quantum Machine Learning Giuseppe Di Molfetta & Hachem Kadri CANA & QARMA , Lab. d’Informatique Fondamentale de Marseille Aix-Marseille Universit´ e, France
Outline I Machine Learning I Quantum Computing I Quantum Machine Learning
Machine Learning? Machine learning is the study of computer algorithms that improve automatically through experience T. Mitchell, 1997 Machine learning is programming computers to optimize a performance criterion using example data or past experience E. Alpaydin, 2004
Annotation/d´ ecodage d’images (de Wolfram, ML Mathematica toolbox) (de Haxby et al, 2001)
AlphaGo (Silver et al. 2016)
Process, Generalization validation interprétation connaissance apprentissage modèles sélection prétraitement données préparées données brutes Goal From a training set consisting of randomly sampled pairs of (input, target), learn a function or a predictor which predicts well the target of a new data. Supervised learning / Generalization ≠ æ Given l training examples ( x 1 , y 1 ) , . . . , ( x l , y l ) œ ( X ◊ Y ) and u test data x l + 1 , . . . , x l + u œ X ≠ æ Learn f : X æ Y to generalize from training to testing
Positionnement V. Vapnik pose, ` a la fin des ann´ ees 70, les bases math´ ematiques de l’apprentissage automatique/statistique, ` a l’intersection de l’informatique, la statistique math´ ematique, l’optimisation.
Perceptron (Rosenblatt, 1958) Inspiration: biological neural network Motivations: I Learning system composed by associating simple processing units I E ffi ciency, scalability, and adaptability Perceptron: a linear classifier, X = R d , Y = { − 1 , +1 } biais : activation = 1 σ w 0 x 1 w 1 σ ( P d i =1 w i x i + w 0 ) x = x 2 w 2
Perceptron (Rosenblatt, 1958) Inspiration: biological neural network Motivations: I Learning system composed by associating simple processing units I E ffi ciency, scalability, and adaptability Perceptron: a linear classifier, X = R d , Y = { − 1 , +1 } I Classifier weights: w œ R d I Classifier prediction: f ( x ) = sign È w , x Í I Question: how to learn w from training data
Perceptron (Rosenblatt, 1958) Inspiration: biological neural network Motivations: I Learning system composed by associating simple processing units I E ffi ciency, scalability, and adaptability Algorithm: S = { ( X n , Y n ) } N n =1 w Ω 0 while it exists ( X n , Y n ): Y n È w , X n Í Æ 0 do w Ω w + Y n X n end while
Perceptron in action
Perceptron in action
Perceptron in action
Perceptron in action
Perceptron: some results Theorem (Number of iterations, Noviko ff , 1962) If it exists γ > 0 , w ∗ , Î w ∗ Î = 1 , Î X n Î Æ R , ’ n = 1 , . . . , N, and Y n È w ∗ , X n Í Ø γ then the number of mistakes made by the Perceptron algorithm is at most R 2 / γ 2 Theorem (XOR, Minsky, Papert, 1969) The perceptron algorithm cannot solve the XOR problem
Neural Networks s j = P σ ( x ) = tanh( x ) i w ji a i +1 a j = σ ( s j ) − 1 w ji w kj s i , a i s k , a k 1 σ ( x ) = 1+exp( − x ) i j k +1 biais : activation = 1 i k x 1 y 1 j x = = y x 2 y 2
SVM and Kernel Methods support vectors outliers w.x + b = 0 margin = 2 / ||w||
Introduction to Quantum Computation Di Molfetta Giuseppe Computer Science Department Aix-Marseille University
Talk Outline Quantum Walks ! Background ! What is Quantum Computation? ! Quantum Algorithms Grover Alg., an introduction O
Background: Classical Computation Input Computation Output 2 + 2 4 Hello.c Hello World! What is the essence of computation?
Classical Computation Theory Church-Turing Thesis: Computation is anything that can be done by a Turing machine. This definition coincides with our intuitive ideas of computation: addition, multiplication, binary logic, etc… Input What is a Turing machine? …0100101101010010110… Finite State Automaton (control module) Infinite Computation tape …0000001011111111100… Read/Write head …0100101101010010110… Output …1110010110100111101…
Classical Computation Theory What kind of systems can perform universal computation? DNA Desktop computers Billiard balls These can all be shown to be equivalent to each other and to a Turing machine! Cellular automata The Big Question: What next?
Talk Outline ! Background ! What is Quantum Computation? ! Quantum Algorithms
What Is Quantum Computation? Conventional computers, no matter how exotic, all obey the laws of classical physics.
What Is Quantum Computation? Conventional computers, no matter how exotic, all obey the laws of classical physics. | α | 2 + | β | 2 = 1 β α + On the other hand, a quantum computer obeys the laws of quantum physics.
The Bit The basic component of a classical computer is the bit, a single binary variable of value 0 or 1. 1 0 At any given time, the value of a bit is either ‘0’ or ‘1’. 0 1 The state of a classical computer is described by some long bit string of 0s and 1s. 0001010110110101000100110101110110...
The Qubit Bit is 1-D point in only one of two states, 0 and 1.
The Qubit Pbit is a 2-D line between the two states 0 and 1 pbit = p ∗ [1] + (1 − p ) ∗ [0]
The Qubit Qubit is a 3-D sphere with 0 and 1 at the poles, and an infinite number of superpositions as points on the sphere | ψ i = cos( θ / 2) | 0 i + e i φ sin( θ / 2) | 1 i
The Qubit | 0 i
The Qubit | 1 i
The Qubit p ( | 0 i � e i π / 4 | 1 i ) / 2
Computation with Qubits How does the use of qubits affect computation? Classical Computation Quantum Computation Data unit: qubit Data unit: bit =|0 � = ‘0’ =|1 � = ‘1’ Valid states: Valid states: | ψ � = c 1 |0 � + c 2 |1 � x = ‘0’ or ‘1’ x = 1 x = 0 | ψ � = |1 � | ψ � = (|0 � + |1 � )/ √ 2 | ψ � = |0 � 0 0 1 1
Computation with Qubits How does the use of qubits affect computation? Classical Computation Quantum Computation Operations: unitary Operations: logical Valid operations: Valid operations: in 1 0 0 1 σ z = σ X = 1 0 0 -1 0 1 1-bit 1-qubit NOT = 1 1 0 i 1 out 1 0 H d = σ y = √ 2 1 -1 -i 0 in 1 0 0 0 0 1 0 1 0 0 2-bit 2-qubit CNOT = AND = 0 0 0 0 0 0 1 in out 1 0 1 0 0 1 0
Computation with Qubits How does the use of qubits affect computation? Classical Computation Quantum Computation Operations: unitary Operations: logical Valid operations: Valid operations: in 1 0 0 1 σ z = σ X = 1 0 0 -1 0 1 1-bit 1-qubit NOT = 1 1 0 i 1 out 1 0 H d = σ y = √ 2 1 -1 -i 0 in 1 0 0 0 0 1 0 1 0 0 2-bit 2-qubit CNOT = AND = 0 0 0 0 0 0 1 in out 1 0 1 0 0 1 0
Computation with Qubits How does the use of qubits affect computation? Classical Computation Quantum Computation Measurement: stochastic Measurement: deterministic Result of measurement State Result of measurement State | ψ � = |0 � ‘0’ x = ‘0’ ‘0’ ‘1’ x = ‘1’ | ψ � = |1 � ‘1’ ‘0’ 50% | ψ � = |0 � - |1 � √ 2 ‘1’ 50%
More than one qubit Two qubits Single qubit |00 � ,|01 � ,|10 � ,|11 � |0 � ,|1 � 0 0 0 1 Hilbert 1 0 1 0 0 0 H 2 = , , , H 2 ⊗ 2 = H 2 ⊗ H 2 = space 0 1 0 0 1 0 , 0 0 1 0 c 1 Arbitrary c 1 c 2 c 1 |00 � + c 2 |01 � + = | Ψ � = | ψ � = c 1 |0 � + c 2 |1 � = c 3 state c 3 |10 � + c 4 |11 � c 2 c 4 c 1 u 11 u 12 u 13 u 14 u 11 u 12 c 1 c 2 u 21 u 22 u 23 u 24 U| Ψ � = Operator U| ψ � = c 3 u 31 u 32 u 33 u 34 u 21 u 22 c 2 c 4 u 41 u 42 u 43 u 44
Quantum Circuit Model Example Circuit Two-qubit One-qubit Measurement operation operation ‘1’ |1 � |0 � σ x |1 � CNOT ‘1’ |0 � |1 � |0 � | 1 i ⌦ | 0 i | 1 i ⌦ | 1 i | 0 i ⌦ | 0 i 0 0 0 1 0 0 0 0 0 0 1 0 1 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0 σ x Ä I = CNOT = 1 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0
Quantum Circuit Model Example Circuit 50% 50% | 0 i + | 1 i | 0 i + | 1 i p ? p ‘1’ ‘0’ σ x 2 2 or CNOT ? ‘1’ ‘0’ |0 � |0 � 0 1 1/ √ 2 1/ √ 2 1/ √ 2 0 0 0 or 0 0 0 0 0 1/ √ 2 1/ √ 2 1 0 1/ √ 2 0 0 Entangled state: Separable state: cannot be written can be written as as tensor product tensor product | Ψ � ≠ | φ � ⊗ | χ � | Ψ � = | φ � ⊗ | χ �
Some Interesting Consequences Reversibility Since quantum mechanics is reversible (dynamics are unitary), quantum computation is reversible. |00000000 � | ψφβπμψ � |00000000 � Quantum Superordinacy All classical quantum computations can be performed by a quantum computer. U No cloning theorem It is impossible to exactly copy an unknown quantum state | ψ � | ψ � | ψ � |0 �
Talk Outline ! Background ! What is Quantum Computation? ! Quantum Algorithms
Quantum Algorithms: What can quantum computers do? ! Grover’s search algorithm ! Quantum Walk search algorithm ! Shor’s Factoring Algorithm
Quantum Algorithms: What can quantum computers do? ! Grover’s search algorithm, an introduction ! Quantum Walk search algorithm ! Shor’s Factoring Algorithm
Recommend
More recommend
