Quantum Computing: An Overview for nonspecialists

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Quantum ComputingAn Overviewfor non-specialists
Financial supports from Kinki Univ., MEXT and
JSPS

• Mikio Nakahara
• Department of Physics
• Research Centre for Quantum Computing
• Kinki University, Japan

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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. Shors Factorization Algorithm

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I. Introduction
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More complicated Example
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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.

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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. Shors Factorization Algorithm

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2. Qubits
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2.1 One Qubit
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Candidates of qubits
Electron, Spin 1/2 Nucleus
Grand State and Excited State of Atom or Ion
Photon
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2.2 Two-Qubit System
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2.3 Multi-qubit systems and entangled states
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2.4 Algorithm Unitary Matrix
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Physical Implementation of U
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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. Shors Factorization Algorithm

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3. Quantum Gates, Quantum Circuit and Quantum
Computer
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3.2 Quantum Gates
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Hadamard transform
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n-qubit Operations
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Quantum Mechanics
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3.3 Universal Quantum Gates
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3.4 Quantum Parallelism and Entanglement
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Power of Entanglement
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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. Shors Factorization Algorithm

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4. Simple Quantum Algorithms4.1 Deutschs
Algorithm
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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. Shors Factorization Algorithm

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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, )

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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.
• The ability to initialize the state of the qubits
  to a simple fiducial state, such as 000gt.
• Long decoherence times, much longer than the gate
  operation time.
• A universal set of quantum gates.
• A qubit-specific measurement capability.

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DiVincenzo 2004_at_Kinki Univ.
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Physical Realization NMR
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Physical Realization Trapped Ions
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Physical Realization Josephson Junction Qubits
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Tunable coupling (interaction on demand)
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Physical Realization Neutral Atoms
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Physical Realization Quantum Dots
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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. Shors Factorization Algorithm

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Difficulty of Prime Number Facotrization

• Factorization of N8902083681874790795683198927209
  1600303613264603794247032637647625631554961638351
  is difficult.
• It is easy, in principle, to show the product of
  p9281013205404131518475902447276973338969 and q
  9591715349237194999547 050068718930514279 is N.
• This fact is used in RSA (Rivest-Shamir-Adleman)
  cryptosystem.

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Factorization algorithm
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Realization using NMR (1535)L. M. K.
Vandersypen et al (Nature 2001)
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NMR molecule and pulse sequence (300 pulses)
perfluorobutadienyl iron complex with the two
13C-labelled inner carbons
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Foolproof realization is discouraging ?
Vartiainen, Niskanen, Nakahara, Salomaa (2004)
Foolproof implementation of the factorization
213 X 7 using Shors algorithm requires at least
22 qubits and approx. 82,000 steps!
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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!

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