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. 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 L R

6. Logical irreversibility fiphysical 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. 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. time H 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. U H H target control Quantum circuit for Controlled NOT.

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

26. Uf f-controlled NOT • f must be implemented reversibly. • quantum circuit readout H H |0> -|1> |0> -|1>

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.