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The brief history of quantum computation

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quant- ph/ 9801027 14 Jan 1998 ' ... Schneider,Wiseman,Munro & GJM, quant-ph/9709042. Feynman-Deutsch principle and measurement. ... – PowerPoint PPT presentation

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Title: The brief history of quantum computation


1
The brief history of quantum computation
  • G.J. Milburn
  • Centre for Laser Science
  • Department of Physics, The University of
    Queensland

2
Outline of talk.
  • The brief history of quantum computation.
  • Deutsch and quantum parallelism.
  • The Shor breakthrough.
  • Physical realisation and future technology.
  • Measurement and computation.
  • Quantum computers and the foundations of physics.

3
Paths to a quantum computer.
  • Two tracks converge to quantum computation
  • R.P. Feynman, 1982
  • Simulating physics with computers,
  • Int. J. Theor. Phys. 21, 467 (1982).
  • R. Landauer, 1961
  • Irreversibility and heat generation in the
    computing process.
  • IBM J. Res. Dev. 5 , 183 (1961)

4
Landauers principle
  • To erase one bit of information we must dissipate
    energy

5
Landauers principle explanation
L
R
  • Is the molecule on L or R ?
  • one bit of information
  • To erase, compress to half volume

6
Logical irreversibility ? physical
irreversibility.
  • The NOT gate is reversible
  • The AND gate is irreversible
  • the AND gate erases information.
  • the AND gate is physically irreversible.

7
Reversible computation.
  • Charles Bennett, IBM, 1973.
  • Logical reversibility of computation,
  • IBM J. Res. Dev. 17, 525 (1973).

8
Reversible gates for universal computation.
  • Fredkin, Toffoli 1979.
  • minimum of three inputs and three outputs
  • eg. Fredkin gate

9
Feynmans question.
  • The second track to quantum computation.
  • R.P. Feynman, 1982
  • Simulating physics with computers,
  • Int. J. Theor. Phys. 21, 467 (1982).
  • Can a quantum system be simulated exactly by a
    universal computer ?
  • NO !

10
Classical simulation transport problem.
  • Simulate Boltzmann equation.
  • R particles on a 1-dim lattice of N sites.
  • note, for fields RO (N)
  • How does the calculation scale with N,R ?

11
Classical probabilistic simulation.
  • Use random numbers to simulate coarse grained
    dynamics.
  • The statistics of random numbers is classical.
  • Cannot simulate a large quantum process.

12
The Feynman processor.
  • A physical computer operating by quantum rules.
  • could it compute more efficiently than a
    classical computer ?

13
Universal computation.
  • Turing machines.
  • See R. Penrose, The Emperors New Mind, page 71.
  • Church-Turing thesis
  • A computable function is one that is computable
    by a universal Turing machine.

14
Computational efficiency.
  • N a number to specify the input to a Turing
    machine.
  • Code as log N bits.
  • Efficient algorithm

15
Deutsch and quantum parallelism.
  • D. Deutsch, 1985
  • Quantum theory, the Church-Turing principle and
    the universal quantum computer.
  • Proc. Roy. Soc. A400, 97, (1985).
  • Feynman-Deutsch principle
  • (Church-Turing principle)
  • Every finitely realisable physical system can be
    perfectly simulated by a universal model
    computing machine operating by finite means

16
Deutsch processor.
  • Computational basis
  • Direct product Hilbert space of N two-level
    systems
  • Quantum Turing machines
  • remain in computational basis state at end of
    each step.
  • Quantum computer
  • arbitrary superpositions of computational
    basis...explore all 2N dimensions !

17
Quantum parallelism.
  • Code binary string for input as an integer.
  • Quantum TM.
  • Quantum parallelism

18
The qubit.
  • A single two-state system can store a single bit
    in computational basis.
  • Superpositions are allowed
  • the qubit.

19
The elementary single qubit operation.
  • The Hadamard transform.
  • Quantum circuit

20
A quantum optical example.
  • A two-state system with a single photon.
  • use a direction qubit

21
Quantum parallel input.
  • prepare an even superposition of all 2N-1
    binary strings.

22
Universal quantum gates.
  • One-qubit gate
  • Hadamard gate
  • Two-qubit gate
  • quantum controlled NOT gate

23
Controlled NOT from spin-spin coupling.
  • Step 1 Hadamard transform of target,
  • Step 2 Spin-spin coupling to control,
  • Step 3 Hadamard transform of target,

24
Quantum circuit for Controlled NOT.
25
Deutschs algorithm.
  • Is f EVEN, f(0) f(1)
  • or ODD, f(0) p f(1) ?
  • Only evaluate f once.

26
f-controlled NOT
  • f must be implemented reversibly.
  • quantum circuit

readout
H
H
0gt -1gt
0gt -1gt
27
Output states at readout.
28
Implementation of Deutsch algorithm.
  • quant- ph/ 9801027 14 Jan 1998
  • Implementation of a Quantum Algorithm to Solve
    Deutsch's Problem on a Nuclear Magnetic Resonance
    Quantum Computer
  • J. A. Jones M. Mosca, Oxford

29
Shor algorithm.
  • Peter Shor, ATT, 1994
  • a quantum algorithm to find prime factors of
    large composites N
  • public key cryptography no longer safe !
  • Key step
  • find the period of the function
  • (x is random, but GCD(x,N)1)

30
Example.
  • Factor 15.
  • Order4
  • Calculate
  • Factors GCD(48,15)3, GCD(50,15)5

31
Quantum factoring
  • Step 1 run algorithm
  • Step 2 readout and discard output

32
Quantum factoring.
  • Step 3 Discrete Fourier transform.
  • strong interference of paths

33
Quantum factoring.
  • Step 4. Readout register.
  • most likely to obtain a number c such that

34
Physical realisations.
  • Ion traps
  • Cirac Zoller 1994, Phys. Rev. Lett, 74,4094.
  • Cavity QED
  • Turchette et al. 1995, Phys. Rev. Lett,75, 4710
  • NMR
  • Gershenfeld Chuang 1997, Science, 275, 350
  • SQUID
  • Rouse et al.,1995 Phys. Rev. Lett, 75, 1614.
  • Quantum dots
  • Loss di Vincenzo, cond-mat/9701055

35
Ion traps
  • Couple lowest centre-of-mass mode to internal
    electronic states of N ions.

36
Quantum computation at UQ
  • New measurement by QC

von Neumann measurement
quantum computation
37
Quantum computation at UQ
  • measure vibrational energy of trapped ions.
  • dHelonGJM Phys. Rev. A. 54, 5141-5146
    (1996).
  • tomographic state reconstruction of vibrational
    state
  • dHelon GJM quant-ph/9705014
  • measurement as a quantum search algorithm
  • Schneider,Wiseman,Munro GJM,
    quant-ph/9709042

38
Feynman-Deutsch principle and measurement.
  • The virtual graduate student part one.

39
Feynman-Deutsch principle and measurement.
  • The virtual graduate student part two.
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