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.

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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)

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Landauers principle

• To erase one bit of information we must dissipate
  energy

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Landauers principle explanation
L
R

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

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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.

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Reversible computation.

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

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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 !

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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 ?

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Classical probabilistic simulation.

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

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The Feynman processor.

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

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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.

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Computational efficiency.

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

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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

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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 !

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Quantum parallelism.

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

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The qubit.

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

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The elementary single qubit operation.

• The Hadamard transform.
• Quantum circuit

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A quantum optical example.

• A two-state system with a single photon.
• use a direction qubit

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Quantum parallel input.

• prepare an even superposition of all 2N-1
  binary strings.

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Universal quantum gates.

• One-qubit gate
• Hadamard gate
• Two-qubit gate
• quantum controlled NOT gate

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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,

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Quantum circuit for Controlled NOT.
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Deutschs algorithm.

• Is f EVEN, f(0) f(1)
• or ODD, f(0) p f(1) ?
• Only evaluate f once.

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f-controlled NOT

• f must be implemented reversibly.

• quantum circuit

readout
H
H
0gt -1gt
0gt -1gt
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Output states at readout.
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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

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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)

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Example.

• Factor 15.

• Order4
• Calculate
• Factors GCD(48,15)3, GCD(50,15)5

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Quantum factoring

• Step 1 run algorithm

• Step 2 readout and discard output

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Quantum factoring.

• Step 3 Discrete Fourier transform.
• strong interference of paths

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Quantum factoring.

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

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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

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Ion traps

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

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Quantum computation at UQ

• New measurement by QC

von Neumann measurement
quantum computation
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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

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Feynman-Deutsch principle and measurement.

• The virtual graduate student part one.

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Feynman-Deutsch principle and measurement.

• The virtual graduate student part two.