Quantum Artificial Intelligence with Qiskit

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
Author
CHAPTER 1: Artificial Intelligence
	1.1. A SHORT HISTORY OF AI
		1.1.1. Cybernetics
		1.1.2. Symbolic Artificial Intelligence
		1.1.3. Connectionist Movement
		1.1.4. Deep Learning
		1.1.5. Quantum Artificial Intelligence
	1.2. SYMBOLICAL ARTIFICIAL INTELLIGENCE
		1.2.1. Bits
		1.2.2. Rules and Operators
		1.2.3. Production Systems
		1.2.4. Tree Search
		1.2.5. Informed Tree Search
	1.3. MACHINE LEARNING
		1.3.1. Vector Representation
		1.3.2. Nearest Neighbor
		1.3.3. Associative Memory
		1.3.4. Artificial Neuron
		1.3.5. Perceptron
		1.3.6. Support Vector Machine
		1.3.7. Support Vector Machine as a Kernel Machine
		1.3.8. Deep Learning
CHAPTER 2: Quantum Physics and Quantum Computation
	2.1. QUANTUM MEASUREMENT
		2.1.1. Interpretations of Quantum Mechanics
	2.2. PRINCIPLES OF QUANTUM COMPUTATION
		2.2.1. Qubits
		2.2.2. Representation
		2.2.3. Linear Operators
	2.3. COMPOUND SYSTEMS
	2.4. MEASUREMENT
	2.5. COMPUTATION WITH ONE QUBIT
	2.6. COMPUTATION WITH M QUBIT
		2.6.1. Matrix Representation of Serial and Parallel Operations
	2.7. ENTANGLEMENT
	2.8. CLONING
	2.9. PHASE KICK-BACK
	2.10. QUANTUM BOOLEAN GATES
CHAPTER 3: Qiskit
	3.1. SOFTWARE DEVELOPMENT KIT
	3.2. INSTALLATION
	3.3. BACKEND SIMULATOR FUNCTIONS
	3.4. COMPATIBILITY
	3.5. EXAMPLE: QUANTUM COIN
		3.5.1. Statevector Evaluation
		3.5.2. Qasm Simulator Evaluation
	3.6. MATRIX REPRESENTATION
	3.7. QUANTUM CIRCUITS
		3.7.1. Un-computing
		3.7.2. General Multi-Controlled X Gate
		3.7.3. OR Operation
	3.8. DEUTSCH ALGORITHM
	3.9. DEUTSCH ALGORITHM ON A REAL QUANTUM COMPUTER
CHAPTER 4: Quantum Gates
	4.1. BOOLEAN ALGEBRA AND THE QUANTUM GATES
		4.1.1. Identity Gate – I
		4.1.2. NOT Gate, Pauli X Gate – X
		4.1.3. Toffoli Gate – ccX
		4.1.4. Controlled NOT Gate – cX
		4.1.5. SWAP Gate – SWAP
		4.1.6. Controlled SWAP Gate – cS
	4.2. GATES FOR ONE QUBIT
		4.2.1. Clifford Gates for One Qubit
		4.2.2. Pauli Y Gate – Y
		4.2.3. Pauli Z Gate – Z
		4.2.4. S Gate – S
		4.2.5. Sdag Gate – S†
	4.3. ROTATION GATES
		4.3.1. T Gate – T
		4.3.2. T† Gate – T†
	4.4. PARAMETERIZED ROTATION GATES
		4.4.1. RX Gate – RX
		4.4.2. RY Gate – RY
		4.4.3. RZ Gate – RZ
		4.4.4. U Gate – U
		4.4.5. Phase Gate – P
	4.5. CONTROLLED U GATES
		4.5.1. Controlled Phase Gate
		4.5.2. Controlled Hadamard Gate – cH
	4.6. UNIVERSALITY
	4.7. QUANTUM CIRCUITS
CHAPTER 5: Grover’s Amplification
	5.1. SEARCH AND QUANTUM ORACLE
		5.1.1. Quantum Oracle
	5.2. HOUSEHOLDER REFLECTION
	5.3. GROVER’S AMPLIFICATION
		5.3.1. Number of Iteration
		5.3.2. Circuit Representation
	5.4. NUMPY EXAMPLE WITH MATRIX NOTATION
	5.5. DECOMPOSITION
	5.6. QISKIT EXAMPLES
	5.7. UN-COMPUTATION
	5.8. GENERALIZATION OF ΛM FOR M QUBITS
CHAPTER 6: SAT Problem
	6.1. FORMULA SATISFIABILITY
	6.2. SAT PROBLEM AND NP COMPLETE
	6.3. SAT PROBLEM AND GROVER’S ALGORITHM
		6.3.1. Quantum Boolean Circuit
		6.3.2. Un-Computation
		6.3.3. Grover’s Amplification
		6.3.4. No Solution
CHAPTER 7: Symbolic State Representation
	7.1. BIT REPRESENTATION OF OBJECTS AND ATTRIBUTES
		7.1.1. “What” and “Where”
	7.2. TREE SEARCH AND THE PATH DESCRIPTORS
	7.3. QUANTUM TREE SEARCH
CHAPTER 8: Quantum Production System
	8.1. PURE PRODUCTION SYSTEMS
		8.1.1. Quantum Production Systems
	8.2. EXAMPLE: SORTING A STRING
		8.2.1. Quantum Production System for Sorting a String
		8.2.2. Number of Iteration
	8.3. COGNITIVE ARCHITECTURE
	8.4. CONTROL FUNCTION
CHAPTER 9: 3 Puzzle
	9.1. 3 PUZZLE
	9.2. REPRESENTATION
		9.2.1. Rules and Trace
	9.3. SEARCH OF DEPTH TWO
	9.4. SEARCH DEPTH THREE
	9.5. SEARCH DEPTH THREE WITH TWO ITERATIONS
CHAPTER 10: 8 Puzzle
	10.1. REPRESENTATION
	10.2. NUMBER OF ITERATIONS
CHAPTER 11: Blocks World
	11.1. REPRESENTATION
		11.1.1. Rules (Productions)
		11.1.2. Oracle
		11.1.3. Architecture
	11.2. EXAMPLES
	11.3. NUMBER OF ITERATIONS
CHAPTER 12: Five Pennies Nim Game
	12.1. QUANTUM CIRCUIT
		12.1.1. Representation of Rules
		12.1.2. Oracle
		12.1.3. Search of Depth Two
	12.2. LIMITATIONS OF QUANTUM TREE SEARCH
CHAPTER 13: Basis Encoding of Binary Vectors
	13.1. BINARY VECTORS
	13.2. SUPERPOSITIONS OF BINARY PATTERNS
		13.2.1. Storage Algorithm
		13.2.2. Qiskit Example
	13.3. ENTANGLEMENT OF BINARY PATTERNS
		13.3.1. Qiskit Example
	13.4. COMPARISON
CHAPTER 14: Quantum Associative Memory
	14.1. QUANTUM NEAREST NEIGHBOR
	14.2. QUANTUM ASSOCIATIVE MEMORY (QuAM)
		14.2.1. Non-Uniform Distribution
		14.2.2. Ventura Martinez Trick
	14.3. INPUT DESTRUCTION PROBLEM
CHAPTER 15: Quantum Lernmatrix
	15.1. LERNMATRIX
		15.1.1. Learning and Retrieval
		15.1.2. Storage Capacity
	15.2. MONTE CARLO LERNMATRIX
	15.3. QUANTUM COUNTING ONES
	15.4. QUANTUM LERNMATRIX
		15.4.1. Generalization
		15.4.2. Example
		15.4.3. Applying Trugenberger Amplification Several Times
		15.4.4. Tree-Like Structures
	15.5. CONCLUSION
CHAPTER 16: Amplitude Encoding
	16.1. AMPLITUDE ENCODING EXAMPLE
	16.2. TOP-DOWN DIVIDE STRATEGY
		16.2.1. Level 1
		16.2.2. Level 2
		16.2.3. Level 3
	16.3. COMBINING STATES
		16.3.1. Level 2
		16.3.2. Level 3
	16.4. QISKIT AMPLITUDE CODING
	16.5. SWAP TEST
		16.5.1. Example for Two-Dimensional Vectors
		16.5.2. Example for Four-Dimensional Vectors
CHAPTER 17: Quantum Kernels
	17.1. QUANTUM KERNELS
	17.2. QUANTUM KERNELS AND SWAP TEST
		17.2.1. Example for Two-Dimensional Vectors
	17.3. QUANTUM KERNELS AND INVERSION TEST
		17.3.1. Example
		17.3.2. Quantum Feature Maps
	17.4. QUANTUM SUPPORT VECTOR MACHINE
CHAPTER 18: qRAM
	18.1. QUANTUM RANDOM ACCESS MEMORY
		18.1.1. The Bucket Brigade Architecture of qRAM
		18.1.2. Amplitude Coding
CHAPTER 19: Quantum Fourier Transform
	19.1. DISCRETE FOURIER TRANSFORM
	19.2. QUANTUM FOURIER TRANSFORM
	19.3. QFT DECOMPOSITION
		19.3.1. QFT for Two qubits
		19.3.2. QFT for Three qubits
		19.3.3. QFT for Four qubits
		19.3.4. QFT Costs
		19.3.5. QFT
CHAPTER 20: Phase Estimation
	20.1. KITAEV’S PHASE ESTIMATION ALGORITHM
		20.1.1. Example with T Gate
	20.2. QUANTUM COUNTING
		20.2.1. Example
CHAPTER 21: Quantum Perceptron
	21.1. COUNTING OF ONES WITH KITAEV’S PHASE ESTIMATION ALGORITHM
	21.2. QUANTUM PERCEPTRON
	21.3. SIMPLE EXAMPLE
CHAPTER 22: HHL
	22.1. QUANTUM ALGORITHM FOR LINEAR SYSTEMS OF EQUATIONS
	22.2. ALGORITHM
		22.2.1. Hamiltonian Simulation
		22.2.2. Kitaev’s Phase Estimation
		22.2.3. Conditioned Rotation
		22.2.4. Un-Computation
		22.2.5. Measurement
		22.2.6. Obtaining the Solution
	22.3. EXAMPLE
		22.3.1. Kitaev’s Phase Estimation to Hamiltonian Simulation
		22.3.2. Conditioned Rotation and Un-Computation
		22.3.3. Obtaining the Solution
	22.4. CONSTRAINTS
CHAPTER 23: Hybrid Approaches − Variational Classification
	23.1. VARIATIONAL CLASSIFICATION
		23.1.1. Example
	23.2. CROSS ENTROPY LOSS FUNCTION
		23.2.1. Multi-Class Loss Function
	23.3. SPSA OPTIMIZER
		23.3.1. Qiskit Variational Quantum Classifier
CHAPTER 24: Conclusion
	24.1. EPILOGUE
	24.2. FURTHER READING
Bibliography
Index