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The brief history of quantum computation

- G.J. Milburn
- Centre for Laser Science
- Department of Physics, The University of

Queensland

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.

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)

Landauers principle

- To erase one bit of information we must dissipate

energy

Landauers principle explanation

L

R

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

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.

Reversible computation.

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

Reversible gates for universal computation.

- Fredkin, Toffoli 1979.
- minimum of three inputs and three outputs
- eg. Fredkin gate

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 !

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 ?

Classical probabilistic simulation.

- Use random numbers to simulate coarse grained

dynamics. - The statistics of random numbers is classical.
- Cannot simulate a large quantum process.

The Feynman processor.

- A physical computer operating by quantum rules.
- could it compute more efficiently than a

classical computer ?

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.

Computational efficiency.

- N a number to specify the input to a Turing

machine. - Code as log N bits.
- Efficient algorithm

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

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 !

Quantum parallelism.

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

The qubit.

- A single two-state system can store a single bit

in computational basis. - Superpositions are allowed
- the qubit.

The elementary single qubit operation.

- The Hadamard transform.
- Quantum circuit

A quantum optical example.

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

Quantum parallel input.

- prepare an even superposition of all 2N-1

binary strings.

Universal quantum gates.

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

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,

Quantum circuit for Controlled NOT.

Deutschs algorithm.

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

f-controlled NOT

- f must be implemented reversibly.

- quantum circuit

readout

H

H

0gt -1gt

0gt -1gt

Output states at readout.

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

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)

Example.

- Factor 15.

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

Quantum factoring

- Step 1 run algorithm

- Step 2 readout and discard output

Quantum factoring.

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

Quantum factoring.

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

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

Ion traps

- Couple lowest centre-of-mass mode to internal

electronic states of N ions.

Quantum computation at UQ

- New measurement by QC

von Neumann measurement

quantum computation

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

Feynman-Deutsch principle and measurement.

- The virtual graduate student part one.

Feynman-Deutsch principle and measurement.

- The virtual graduate student part two.