Chapter
Encoding and decoding quantum error-correcting codes
Fighting decoherence using entanglement
Encoding stabilizer codes
Example: encoding the five-qubit code
First step: X-only generator
Second step: X-generator of weight one
Third step: row operations
Linear optics quantum computation: an overview
Quantum information processing with linear optics
Quantum optics and quantum information
Classical electromagnetic field
Minimum uncertainty states
Previous suggestions with optics
Quantum optical Fredkin gate
Cavity quantum electrodynamics
Progress with linear optics
Decomposition of unitaries
Optical simulation of quantum logic
Linear optics quantum computation
Nonlinear sign shift gate
Teleporting qubits through a gate
Teleporting with the C-Sign entangled states
Basic teleportation with linear optics
Increasing the probability of success
Generalised beam splitter
Bounds on success probabilities
LOQC and quantum error correction
Improving LOQC: beyond state preparation
Quantum error correcting codes
Z-measurement quantum error correcting code (QECC)
Properties of the Z-measurement QECC
Threshold for Z-measurement QECC
Accuracy threshold theorem
Teleportation with error recovery
Entanglement in quantum optics
A general separability criterion
Relation with other criteria
Continuous variable systems
Phase-space representations
Continuous variable entanglement
CV tripartite entanglement
Entanglement in graph states and its applications
Definitions for graph states
Stabilizer states and codes
Local Clifford group and LC equivalence
Generalizations to d-level systems
Remarks on harmonic systems
Clifford operations and classical simulation
Examples and applications
Cluster states and the one-way quantum computer
Quantum error correcting codes
CSS states and secret sharing
Entanglement purification and secure state distribution
Reduced states of graph states
Equivalence classes under local operations
Entanglement in graph states
Bell inequalities and entanglement witnesses
Two-particle correlations and localizable entanglement
Graph states in the presence of decoherence
Stability of entanglement
Entanglement purification
Multipartite secure state distribution
Quantum algorithms and quantum chaos
Classical and quantum chaos
Many-body quantum chaos: application to quantum computers
Quantum chaos in many-body interacting systems
Quantum chaos in quantum computers hardware
Emergence of quantum chaos in quantum computers
Effects of quantum chaos on quantum computers
Introduction to quantum algorithms
Quantum algorithms for quantum chaotic maps
Quantum simulation of quantum maps
Kicked rotator and sawtooth map
Other maps displaying specific physical phenomena
Kicked Harper model: three possible algorithms
Direct measurement of wave function
Wigner and Husimi distributions
Quantum simulation of classical chaos
Simulation of classical maps
Dissipative maps and strange attractors
Extraction of information
Fourier coefficients, correlation functions
Recurrence times and periodic orbits
Quantum chaos, decoherence and quantum computation
Remarks on classical and quantum chaos
Time scales of quantum chaos
Dynamical stability of quantum motion
The quantum Loschmidt echo
Effects of imperfections in the quantum computer hardware
The quantum sawtooth map model
Quantum vs. classical errors
Static imperfections vs. noisy gates
On the stability of local and non-local characteristics
Quantum noise and quantum trajectories
An example from quantum optics: spontaneous emission
Generalized amplitude damping
Electron-hole entanglement in the Fermi sea
Quantum vs. classical correlations
Entanglement measures for pure states
Entanglement measures for mixed states
How to entangle free particles
Spin vs. orbital entanglement
Entanglement detection by noise measurements
Beyond the tunneling regime
Loss of entanglement by dephasing
Quantum entanglement pump
Teleportation by electron-hole annihilation
The experimental challenge
Appendix A. Entanglement production for spin-dependent scattering
Appendix B. Entanglement production at finite temperature
Appendix C. Bell inequality with noise correlators
Entanglement and matrix product states in one-dimensional quantum lattice systems
Entanglement in one-dimensional quantum systems
Efficient simulation of time evolution in one-dimensional quantum many-body systems
Beyond matrix product states
Mesoscopic quantum measurements
Measurements dynamics of ballistic mesoscopic detectors
Back-action dephasing rate
Information acquisition rate
Tunneling without tunneling: wave function reduction in a mesoscopic qubit
Applications of quantum filtering and feedback
Classical filtering and feedback control
Quantum filtering and feedback
Applications of quantum filtering and feedback
Adaptive homodyne measurement of optical phase
Broadband magnetometry with atoms
Nanophotonic signal processing via cavity QED with strong coupling
Quantum computation with Josephson qubits
Single Josephson junction qubits
Decoherence in superconducting qubits
Mechanisms and models of decoherence
Electromagnetic environment
Effect of classical noise
Combining noise from different classes
Geometric quantum computation
Geometric phases in superconducting nanocircuits
Geometric control phase shift
Geometric phases in open systems
Holonomic quantum computation with Josephson circuits
Stimulated Raman adiabatic passage in superconducting nanocircuits
Connection to quantum pumping
Quantum state transmission
Solid-state quantum bit circuits
Why solid-state quantum bit circuits?
Criteria required for qubits
Qubit implementation: Atoms and ions vs. electrical circuits
Solid-state electrical qubit circuits
Qubits based on semiconductor structures
Kane's proposal: nuclear spins of P impurities in silicon
Charge states in quantum dots
Electron spins in quantum dots
Superconducting qubit circuits
Hamiltonian of Josephson circuits
How to maintain quantum coherence?
Qubit-environment coupling Hamiltonian
Decoherence = relaxation + dephasing
The optimal working point strategy
Relaxation and dephasing in the quantronium
Readout of the quantronium
AC methods for QND readout
Coherent control of the qubit
NMR-like control of a qubit
Decoherence during free evolution
Decoherence during driven evolution
First experimental results
Tunable vs. fixed couplings
Control of the interaction mediated by a fixed Hamiltonian
Conclusions and perspectives
Electron spin qubits in quantum dots
Transport though quantum dots
Few-electron quantum dots with integrated charge read-out
Few-electron quantum dots
Quantum point contact as charge detector
Double-dot charge stability diagram
Real-time detection of single-electron tunnelling using a quantum point contact
Single-shot read-out of an individual electron spin in a quantum dot
Measuring electron spin in quantum dots
Two-level pulse technique
Single-shot read-out of one electron spin
Coherent single-spin manipulation: ESR
Coherent spin interactions: SWAP
Quantum computation with trapped ions
Initial state preparation
Addressing individual ions in a string
Outlook: qubit interfacing
Engineering multi-particle entanglement with neutral atoms in optical lattices
Optical lattice potentials
Spin-dependent optical lattice potentials
Bose-Hubbard model of interacting bosons in optical lattices
Ground states of the Bose-Hubbard Hamiltonian
Superfluid-to-Mott-insulator transition
Collapse and revival of a macroscopic quantum field
Quantum gate arrays via controlled collisions
Using controlled collisional quantum gates
Entanglement generation via spin changing collisions
Quantum noise correlations
Classical and quantum dynamics with ultra-cold atoms in billiards
Atom-optic billiards: basic concepts
Chaotic and integrable dynamics
Elliptical billiard and the effect of scattering by impurities
Billiards with soft walls
Billiards with curved trajectories
Microwave spectroscopy of optically trapped atoms
Quantum dynamics in Gaussian trap
Quantum dynamics in mixed and chaotic billiards
Quantum Computers, Algorithms and Chaos
( International School of Physics “Enrico Fermi” )
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