Ebook: Quantum Computers, Algorithms and Chaos

During the last ten years Quantum Information Processing and Communication (QIPC) has established itself as one of the new hot topic fields in physics, with the potential to revolutionize many areas of science and technology. QIPC replaces the laws of classical physics applied to computation and communication with the more fundamental laws of quantum mechanics. This becomes increasingly important due to technological progress reaching smaller and smaller scales where quantum effects start to be dominant. In addition to its fundamental nature, QIPC promises to advance computing power beyond the capabilities of any classical computer, to guarantee secure communication and establish direct links to emerging quantum technologies, such as, for example, quantum-based sensors and clocks.

One of the outstanding features of QIPC is its interdisciplinary character: it brings together researchers from physics, mathematics and computer science. In particular, within physics we have seen the emergence of a new QIPC community, which ranges from theoretical to experimental physics, and crosses boundaries of traditionally separated disciplines such as atomic physics, quantum optics, statistical mechanics and solid-state physics, all working on different and complementary aspects of QIPC.

In the spirit of the interdisciplinary character of QIPC, the purpose of the School was to bring together world leading experts to give lectures on the foundations of QIPC, and on theoretical and experimental questions of QIPC implementations with different physical devices. The School covered the following topics:

- Introduction to quantum computing.

- Quantum logic, information and entanglement.

- Quantum algorithms. Error-correcting codes for quantum computations.

- Quantum measurements and control.

- Quantum communication.

- Quantum optics and cold atoms for quantum information.

- Quantum computing with solid-state devices.

- Theory and experiments for superconducting qubits.

- Interactions in many-body systems: quantum chaos, disorder and random matrices.

- Decoherence effects for quantum computing.

- Future prospects of quantum information processing.

The School attracted a large number of applications from all over the world and attained its maximum capacities of nearly a hundred participants. This clearly shows the great interest of young researchers in the field of QIPC. To a good extent this is due to the recent impressive experimental progress achieved with various physical implementations of quantum information processors, highlighted in the lecture courses given during the School. They include ion-trap–based quantum computers, Josephson junction qubits, semiconductor quantum dots, cold atoms and optical lattices, linear optics quantum computation and entangled photons. The School also highlighted the deepening in our understanding of theoretical aspects of quantum computation and quantum communication, including topics like quantum error-correcting codes, quantum algorithms for complex dynamics, quantum measurements and feedback control, decoherence and imperfections effects for the accuracy of computation, applications of quantum chaos to systems with many qubits, entanglement in mesoscopic structures, critical phenomena and one-way quantum computation.

With the rapid development of QIPC we are witnessing the emergence of a new field in physics, mathematics and computer science. The enthusiasm, which is generated by this new field in the physics community, was clearly visible at the School, with young promising people entering this new field, for whom QIPC will be a major part in their future scientific careers.

We acknowledge financial support provided to the School and students by Università degli Studi dell’Insubria, the FET-QIPC programme of European Union, UNESCO-ROSTE and U.S. Army European Research Office. The School benefited from the nice environment and general atmosphere and from the support of the staff of the Italian Physical Society, which we gratefully acknowledge.

G. Casati, D. L. Shepelyansky, P. Zoller and G. Benenti

1. Introduction

2. Three bit code

3. Binary fields and discrete vector spaces

4. Classical error correction

5. Quantum erro correction

6. Code construction

7. Further insights into coding and syndrome extraction

8. The physics noise

1. Introduction

2. Fighting decoherence using entanglement

3. Correcting errors

4. Encoding stabilizer codes

1. Quantum information processing with linear optics

2. Linear optics quantum computation

3. LOQC and quantum error correction

4. Conclusion

1. Introduction

2. A general separability criterion

3. Relation with other criteria

4. Continuous variable systems

5. Phase-space representations

6. Continuous variable entanglement

7. CV tripartite entanglement

1. Introduction

2. Definitions for graph states

3. Clifford operations and classical simulation

4. Examples and applications

5. Physical implementations

6. Reduced states of graph states

7. Equivalence classes under local operations

8. Entanglement in graph states

9. Weighted graph states

10. Graph states in the presence of decoherence

11. Summary

1. Introduction

2. Classical and quantum chaos

3. Many-body quantum chaos: application to quantum computers

4. Introduction to quantum algorithms

5. Quantum algorithms for quantum chaotic maps

6. Quantum simulation of classical chaos

7. General conclusion

1. Introduction

2. Remarks on classical and quantum chaos

3. Effects of imperfections in the quantim computer hardware

4. Quantum noise and quantum trajectories

5. Final remarks

1. Introduciton

2. Entanglement basics

3. How to entangle free particles

4. Spin vs. orbital entanglement

5. Entanglement detection by noise measurements

6. Loss of entanglement by dephasing

7. Quantum entanglement pump

8. Teleportation by electron-hole annihilation

9. Three-qubit entanglement

10. The experimental challenge

Introduction

Matrix product states

Entanglement in one-dimensional quantum systems

Efficient simulation of time evolution in one-dimensional quantum many-body systems

Beyond matrix product states

1. Introduction

2. Measurements dynamics of ballistic mesoscopic detectors

3. Tunneling without tunneling: wave function reduction in a mesoscopic qubit

4. Tunneling detectors

5. Conclusion

1. Classical filtering and feedback control

2. Quantum filtering and feedback

3. Applications of quantum filtering and feedback

4. Continuing research

1. Introduction

2. Josephson qubits

3. Decoherence in superconducting qubits

4. Geometric quantum computation

5. Few qubits applications

1. Why solid-state quantum bit circuits?

2. Towards quantum machines

3. Qubits based on semiconductor structures

4. Superconducting qubit circuits

5. The quantronium circuit

6. Coherent control of the qubit

7. Probing qubit coherence

8. Qubit coupling schemes

9. Conclusions and perspectives

1. Introduction

2. Few-electron quantum dots with integrated charge read-out

3. Real-time detection of single-electron tunnelling using a quantum point contact

4. Single-shot read-out of an individual electron spin in quantum dot

5. Coherent control

6. Outlook

1. Introduction

2. Ion storage

3. Laser interaction

4. Experimental techniques

5. Recent progress

6. New methods

7. Outlook: qubit interfacing

1. Introduction

2. Optical lattices

3. Bose-Hubbard model of interacting bosons in optical lattices

4. Collapse and revival of a macroscopic quantum field

5. Quantum gate arrays via controlled collisions

6. Entanglement generation via spin changing collisions

7. Quantum noise correlations

8. Outlook

1. Introduction

2. Atom-optic billiards: basic concepts

3. Classical dynamics

4. Quantum dynamics

5. Summary