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. Landauer’s principle To erase one bit of information we must dissipate energy

5. Landauer’s principle: explanation Is the molecule on L or R ? –one bit of information To erase, compress to half volume LR

6. Logical irreversibility fi 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. Feynman’s 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. R particles on a 1-dim lattice of N sites. –note, for fields R= O (N) How does the calculation scale with N,R ? Simulate Boltzmann equation.

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 Emperor’s 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. A 400, 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 2 N 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: H time

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 2 N-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. HH U control target

25. Deutsch’s algorithm. Is f EVEN, f(0) = f(1) or ODD, f(0) π f(1) ? Only evaluate f once.

26. f-controlled NOT f must be implemented reversibly. quantum circuit HH UfUf |0> -|1> readout

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, AT&T, 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. Order=4 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.  d’Helon&GJM Phys. Rev. A. 54, 5141-5146 (1996). tomographic state reconstruction of vibrational state  d’Helon & 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.