Quantum Computing: An Overview for non-specialists Mikio Nakahara Department of Physics & Research Centre for Quantum Computing Kinki University, Japan Financial supports from Kinki Univ., MEXT and JSPS

Tehran 2009 Plan of lecture 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran 2009 I. Introduction

Tehran 2009 More complicated Example

Tehran 2009 Quantum Computing/Information Processing Quantum computation & information processing make use of quantum systems to store and process information. Exponentially fast computation, totally safe cryptosystem, teleporting a quantum state are possible by making use of states & operations which do not exist in the classical world.

Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran Qubits

Tehran One Qubit

Tehran 2009 Candidates of qubits ： Electron, Spin 1/2 Nucleus Photon Grand State and Excited State of Atom or Ion

Tehran Two-Qubit System

Tehran Multi-qubit systems and entangled states

Tehran Algorithm = Unitary Matrix

Tehran 2009 Physical Implementation of U

Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran Quantum Gates, Quantum Circuit and Quantum Computer

Tehran 2009

3.2 Quantum Gates

Tehran 2009 Hadamard transform

Tehran 2009

n-qubit Operations

Tehran 2009 Quantum Mechanics

Tehran Universal Quantum Gates

Tehran Quantum Parallelism and Entanglement

Tehran 2009 Power of Entanglement

Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran Simple Quantum Algorithms 4.1 Deutsch’s Algorithm

Tehran 2009

Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran 2009 Necessary Conditions for a PC to Work Properly Hardware (Memory, CPU etc), Able to reset all the memories to 0, The PC lasts till a computation stops (maybe a problem if it takes more than 10 years to finish the computation.) Able to carry out any logic operations Able to output the results (display, printer, …)

Tehran 2009 Necessary Conditions for a Quantum Computer to Work Properly (DiVincenzo Criteria) Hardware (Memory, CPU etc) Able to reset all the memories to 0, The PC lasts till a computation stops. Able to carry out any logic operations Able to output the results (display, printer, ) A scalable physical system with well characterized qubits. A scalable physical system with well characterized qubits. The ability to initialize the state of the qubits to a simple fiducial state, such as |00…0>. The ability to initialize the state of the qubits to a simple fiducial state, such as |00…0>. Long decoherence times, much longer than the gate operation time. Long decoherence times, much longer than the gate operation time. A “universal” set of quantum gates. A “universal” set of quantum gates. A qubit-specific measurement capability. A qubit-specific measurement capability.

Tehran 2009 DiVincenzo Univ.

Tehran 2009 Physical Realization: NMR

Tehran 2009 Physical Realization: Trapped Ions

Tehran 2009 Physical Realization: Josephson Junction Qubits

Tehran 2009 Tunable coupling (interaction on demand)

Tehran 2009 Physical Realization: Neutral Atoms

Tehran 2009 Physical Realization: Quantum Dots

Tehran 2009 Plan of lectures 1. Introduction 2. Qubits 3. Quantum Gates, Quantum Circuits and Quantum Computer 4. Simple Quantum Algorithms 5. DiVincenzo Criteria & Physical Realizations 6. Shor’s Factorization Algorithm

Tehran 2009 Difficulty of Prime Number Facotrization Factorization of N= is difficult. It is easy, in principle, to show the product of p= and q = is N. This fact is used in RSA (Rivest-Shamir- Adleman) cryptosystem.

Tehran 2009 Factorization algorithm

Tehran 2009 Realization using NMR (15=3×5) L. M. K. Vandersypen et al (Nature 2001)

Tehran 2009 NMR molecule and pulse sequence (~300 pulses) perfluorobutadienyl iron complex with the two 13C-labelled inner carbons

Tehran 2009

Foolproof realization is discouraging … ? Vartiainen, Niskanen, Nakahara, Salomaa (2004) Foolproof implementation of the factorization 21=3 X 7 using Shor’s algorithm requires at least 22 qubits and approx. 82,000 steps!

Tehran 2009 Summary Quantum information and computation are interesting field to study. (Job opportunities at industry/academia/military). It is a new branch of science and technology covering physics, mathematics, information science, chemistry and more. Thank you very much for your attention!

Tehran 2009