Quantum Optics and Quantum Computation: An introduction

PRELIMS.pdf
Preface
Acknowledgments
Author biographies
Dipankar Bhattacharyya
Jyotirmoy Guha
CH001.pdf
Chapter 1 Bra ket algebra of Dirac
1.1 The bra and ket notation of Dirac
1.2 Hermitian conjugation
1.3 Definition of inner product (also called overlap)
1.4 Definition of outer product
1.5 Eigenvalue equation
1.6 Linear vector space
1.7 Linear independence
1.8 Linear dependence
1.9 Span (expansion of an arbitrary ket)/expansion postulate
1.10 Ket space, bra space, dual space
1.11 Physical significance of inner product 1.12 Norm and the process of normalization
1.13 Ortho-normalization (orthogonal + normalized)
1.14 Orthonormal basis (orthogonal + normalized + linearly independent + span)
1.15 Expansion postulate
1.16 Projection operator
1.17 Normal matrix
1.18 Spectral theorem
1.19 Elements of a matrix in Bra Ket notation
1.20 Hermitian matrix operator
1.21 Unitary matrix
1.22 Diagonalization of a matrix—change of basis
1.23 Triangle laws (inequality and equality)
1.24 Cauchy–Schwarz laws (inequality and equality)
1.25 Commutator bracket
1.26 Trace
1.27 Pauli spin matrices
1.28 Orthogonal matrix operator
1.29 Standard method of ortho-normalization Graham–Schmidt ortho-normalization procedure
1.30 Definition of average value
1.31 Some definitions
1.32 Kroneckar product (symbol ⊗) or direct product or tensor product
1.33 Further reading
1.34 Problems
CH002.pdf
Chapter 2 Postulates of quantum mechanics
2.1 First postulate: observables are replaced by operators
2.2 Second postulate: state vector and wave function
2.3 Third postulate: process of measurement
2.4 Fourth postulate: Time evolution of a state
2.5 Solution of the Schrödinger equation
2.6 Unitary operator keeps the length of state vector constant
2.7 Heisenberg’s uncertainty principle or principle of indeterminism
2.8 Further reading
2.9 Problems
CH003.pdf
Chapter 3 Introduction to quantum computing
3.1 Introduction
3.2 Some basic ideas about classical and quantum computing
3.3 Definition of certain terms relating to quantum computing
3.3.1 Information
3.3.2 Computability
3.3.3 Algorithm
3.3.4 Quantum parallelism
3.3.5 Quantum reversible computing
3.3.6 Ancilas (Ancila bits)
3.3.7 No-cloning theorem
3.3.8 Quantum entanglement
3.3.9 Quantum teleportation
3.3.10 Quantum cryptography
3.3.11 Protocols of quantum computing
3.3.12 Quantum de-coherence
3.4 Journey towards quantum computing
3.4.1 Moore’s law (1965)
3.4.2 Feynman idea of quantum computer (1982)
3.4.3 Deutsch algorithm
3.4.4 Shor’s algorithm
3.4.5 Concept of qubit
3.4.6 Grover’s algorithm
3.4.7 Experimental demonstration of quantum teleportation
3.4.8 First quantum computer
3.4.9 Other milestones
3.5 Need for quantum computers
3.6 Landauer’s principle
3.7 Quantum computing
3.8 Bits 0 and 1
3.9 A bit of Boolean algebra
3.10 Gate
3.11 Computational complexity
3.12 Further reading
3.13 Problems
CH004.pdf
Chapter 4 Quantum bits
4.1 Qubits and comparison with classical bits
4.2 Qubit model applied to the Stern–Gerlach experiment
4.3 Qubit model applied to polarized photon (computational and Hadamard basis introduced)
4.4 Bloch sphere representation of a qubit
4.5 Multiple qubits
4.6 Explicit representation of the basis states
4.7 Bell state or EPR pair (or state)
4.8 Global phase and relative phase
4.9 Measurement depends on choice of basis
4.10 Further reading
4.11 Problems
CH005.pdf
Chapter 5 Quantum circuits
5.1 Quantum gate and quantum circuit
5.2 Single-qubit gates
5.3 Quantum NOT gate or Pauli Xˆ gate (σˆx)
5.4 Zˆ gate or Pauli Zˆ gate (σˆz)
5.5 Pauli Yˆ gate or σˆy
5.6 Phase shift gates (Pˆ gate, Sˆ gate, Tˆ gate)
5.6.1 Pˆ gate
5.6.2 Sˆ gate
5.6.3 Tˆ gate or π8 gate
5.6.4 Zˆ gate as a phase shift gate
5.6.5 Special Pˆ(ϕ) gate: Rˆk=Pˆ(2π 2k)
5.6.6 Inter-relations
5.7 Hadamard gate Hˆ, Hadamard basis ∣+>,∣−>
5.8 Unitary matrix as length preserving matrix
5.9 Rotation gates RˆX(θ),RˆY(θ),RˆZ(θ)
5.9.1 RˆZ(ξ) represents rotation of the Bloch vector about Z-axis by ξ
5.9.2 Rotation about an arbitrary axis (nˆ)
5.9.3 An arbitrary single qubit unitary operator can be converted into a Hadamard gate
5.9.4 RˆX(π)RˆY(π 2) is a Hadamard operator
5.9.5 Evaluation of XˆYˆXˆ,XˆRˆY(θ)Xˆ
5.9.6 Hadamard operation is equivalent to rotation on Bloch sphere about Y-axis by 90° followed by rotation about X-axis by 180°
5.10 Multi-qubit gates
5.11 Controlled-NOT gate or CNOT gate
5.11.1 CNOT gate is Hermitian
5.12 Preparing Bell states
5.13 Swap gate
5.13.1 Construction of SWAP gate using three CNOT gates
5.14 Controlled U gates
5.14.1 Note on controlled Xˆ,Yˆ,Zˆ gates
5.15 Toffoli quantum gate or CCNOT gate (controlled controlled NOT gate)
5.16 Controlled SWAP gate or CS gate or Fredkin gate
5.17 Deutsch gate
5.18 Implementing classical computation by quantum gates
5.18.1 Implementation of NOT operation by quantum gate
5.18.2 Implementation of XOR operation
5.18.3 Implementation of AND operation
5.19 Plan of a quantum circuit
5.20 Quantum half adder circuit
5.21 Quantum full adder circuit
5.22 Oracle (black box) in quantum computer
5.23 Hadamard transformation on each of n qubits leads to a linear superposition of 2n states
5.24 Process of measurement
5.25 Quantum coin flipping
5.26 Further reading
5.27 Problems
CH006.pdf
Chapter 6 Teleportation and super dense coding
6.1 Quantum no-cloning theorem
6.2 Teleportation
6.3 Super dense coding (or dense coding) (of Bennett and Wiesner)
6.4 Further reading
6.5 Problems
CH007.pdf
Chapter 7 Pure and mixed state
7.1 Pure state
7.2 Mixed state
7.3 Density operator (introduced by Von Neumann)
7.4 Density operator for a pure state
7.4.1 Coherence
7.4.2 Decoherence
7.5 Average
7.6 Density operator of a mixed state (or an ensemble)
7.7 Quantum mechanics of an ensemble
7.8 Density matrix for a two-level spin system (Stern–Gerlach experiment)
7.9 Single-qubit density operator in terms of Pauli matrices
7.10 Some illustration of density matrix for pure and mixed states
7.11 Partially mixed, completely mixed, maximally mixed states
7.12 Time evolution of density matrix: Liouville–Von Neumann equation
7.13 Partial trace and the reduced density matrix
7.14 Measurement theory of mixed states
7.15 Positive Operator Valued Measure (POVM)
7.16 Further reading
7.17 Problems
CH008.pdf
Chapter 8 Quantum algorithms
8.1 Quantum parallelism
8.2 Reversibility
8.3 XOR is addition modulo 2
8.4 Quantum arithmetic and function evaluations
8.5 Deutsch algorithm
8.6 Deutsch–Jozsa (DJ) algorithm
8.7 Bernstein–Vazirani algorithm
8.8 Simon algorithm
8.9 Grover’s search algorithm
8.10 Discrete integral transform
8.11 Quantum Fourier transform
8.12 Finding period using QFT
8.13 Implementation of QFT
8.14 Some definitions and GCD evaluation
8.15 Inverse modulo
8.16 Shor’s algorithm
8.17 Further reading
8.18 Problems
CH009.pdf
Chapter 9 Quantum error correction
9.1 Error in classical computing
9.2 Errors in quantum computing/communication
9.3 The phase flip
9.4 Qubit transmission from Alice to Bob
9.5 Converting a phase flip error to qubit flip error
9.6 Shor’s nine-qubit error code
9.6.1 Encoding circuit
9.6.2 Why is it called Shor’s 9-qubit error code?
9.6.3 Decoding circuit
9.7 Further reading
9.8 Problems
CH010.pdf
Chapter 10 Quantum information
10.1 Classical information theory
10.2 Decision tree
10.3 Measure of information: Shannon’s entropy
10.4 Statistical entropy and Shannon’s information entropy
10.5 Communication system
10.6 Shannon’s noiseless coding theorem
10.7 Prefix code, binary tree
10.8 Quantum information theory, Von Neumann entropy
10.9 Further reading
10.10 Problems
CH011.pdf
Chapter 11 EPR paradox and Bell inequalities
11.1 EPR paradox
11.2 David Bohm’s version of EPR paradox (1951)
11.3 Bell’s (Gedanken) experiment: EPR and Bell’s inequalities
11.4 Clauser, Horne, Shimony and Holt’s inequality
11.5 Further reading
11.6 Problems
CH012.pdf
Chapter 12 Cryptography—the art of coding
12.1 A bit of history of cryptography
12.2 Essential elements of cryptography
12.3 One-time pad
12.4 RSA cryptosystem
12.5 Fermat’s little theorem
12.6 Euler theorem
12.7 Chinese remainder theorem
12.8 RSA algorithm
12.9 Quantum cryptography
12.10 Protocol of quantum cryptography
12.10.1 BB84 protocol
12.10.2 B92 protocol
12.10.3 Ekert protocol using EPR pairs (E91)
12.11 Further reading
12.12 Problems
CH013.pdf
Chapter 13 Experimental aspects of quantum computing
13.1 Basic principle of nuclear magnetic resonance quantum computing
13.2 Further reading
CH014.pdf
Chapter 14 Light–matter interactions
14.1 Interaction Hamiltonian
14.2 Rabi oscillations
14.3 Weak field case
14.4 Strong field case: Rabi oscillations
14.5 Damping phenomena
14.6 The density matrix
14.7 Pure and mixed states
14.8 Equation of motion of the density operator
14.9 Inclusion of decay phenomena
14.10 Vector model of density matrix equations of motion
14.11 Power broadening and saturation of the spectrum
14.12 Spectral line broadening mechanism
14.13 Natural broadening
14.14 Collision or pressure broadening
14.15 Inhomogeneous broadening or Doppler broadening
14.16 Further reading
14.17 Problems
CH015.pdf
Chapter 15 Laser spectroscopy and atomic coherence
15.1 Moving two-level atoms in a travelling wave field
15.2 Moving atoms in a standing wave
15.3 Lamb dip
15.4 Crossover resonances
15.5 Atomic coherence phenomena
15.6 EIT Hamiltonian of the system
15.7 Dressed states picture
15.8 Coherent population trapping
15.9 Electromagnetically induced absorption (EIA)
15.10 Further reading
15.11 Problems
CH016.pdf
Chapter 16 Quantum theory of radiation
16.1 Maxwell’s equations
16.2 The electromagnetic field in a cavity
16.3 Quantization of a single mode
16.4 Multimode radiation field
16.5 Coherent states
16.6 Squeezed states of light
16.7 Further reading
16.8 Problems
CH017.pdf
Chapter 17 Interaction of an atom with a quantized field
17.1 Interaction Hamiltonian in terms of Pauli operators
17.2 Absorption and emission phenomena
17.3 Dressed states
17.4 Jaynes–Cummings model
17.5 Theory of spontaneous emission: Wigner–Weisskopf model
17.6 Further reading
17.7 Problems
CH018.pdf
Chapter 18 Photon statistics
18.1 Young’s double-slit experiment
18.2 Hanbury Brown–Twiss experiment
18.3 Photon counter
18.4 Outcome of the photon counter
18.5 Photon statistics of a perfectly coherent light
18.6 Photon statistics of a thermal light
18.7 Classification of light by second-order correlation function and photon statistics.
18.8 Photon bunching and anti-bunching
18.8.1 Coherent light
18.8.2 Bunched light
18.8.3 Anti-bunched light
18.9 Further reading
18.10 Problems