ABSTRACT
It appears, in principle, that the laws of quantum mechanics allow a quantum computer to solve certain mathematical problems more rapidly than can be done using a classical computer. However, in order to build such a quantum computer a number of technological problems need to be overcome. A stepping stone to this goal is the implementation of relatively simple quantum algorithms using current experimental techniques.This paper explores small scale quantum algorithms from two different perspectives. Firstly, it will be shown how small scale quantum algorithms can be tailored to fit current schemes for implementing a quantum computer. Secondly, I will review a simple model of computation, based on read-only-memory. This model allows the comparison of the space-efficiency of reversible error-free classical computation with reversible, error-free quantum computation. The quantum model has been shown to be more powerful than the classical model.
- D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani. Quantum walks on graphs, 2000. quant-ph/ 0012090.Google Scholar
- A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. A. Smolin, and H. Weinfurter. Elementary gates for quantum computation. Physical Review A, 52(5):3457, 1995.Google ScholarCross Ref
- J. C. Bergquist, R. G. Hulet, W. M. Itano, and D. J. Wineland. Observation of quantum jumps in a single atom. Physical Review Letters, 57:1699, 1986.Google ScholarCross Ref
- R. Blatt. Laser cooling of trapped ions. In J. Dalibar, J. Raimond, and J. Zinn-Justin, editors, Fundamental Systems in Quantum Optics, Les Houches, 1990. Elsevier Science Publishers.Google Scholar
- J. I. Cirac and P. Zoller. Quantum computations with cold trapped ions. Physical Review Letters, 74(20):4091, 1995.Google ScholarCross Ref
- C. D'Helon and G. J. Milburn. Measurements on trapped laser-cooled ions using quantum computations. Physical Review A, 54(6):5141, 1996.Google ScholarCross Ref
- F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland. Laser cooling to the zero point energy of motion. Physical Review Letters, 62:403, 1989.Google ScholarCross Ref
- Fortschr Phys. Special issues on quantum computation, 4-8 46 (1998) and 9-11 48 (2000).Google Scholar
- P. K. Ghosh. Ion Traps. Clarendon Press, Oxford, 1995.Google Scholar
- L. K. Grover. Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters, 79(2):325, 1997.Google ScholarCross Ref
- R. J. Hughes, D. F. V. James, J. J. Gomez, M. S. Gulley, M. H. Holzscheiter, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, V. D. Sandberg, M. M. Schauer, C. M. Simmons, C. E. Thorburn, D. Tupa, P. Z. Wang, and A. G. White. The los alamos trapped ion quantum computer experiment. Fortschr. Phys., 46:329--362, 1998.Google ScholarCross Ref
- D. F. V. James. Quantum dynamics of cold trapped ions, with applications to quantum computation. Applied Phyics B, 66:181--190, 1998.Google ScholarCross Ref
- C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould. Resolved-sideband raman cooling of a bound atom to the 3D zero-point energy. Physical Review Letters, 75(22):4011, 1995.Google ScholarCross Ref
- C. Monroe, D. M. Meekhof, B. E. King, and D. J. Wineland. A "Schrödinger cat" superposition state of an atom. Science, 272:1131, 1996.Google ScholarCross Ref
- W. Nagourney, J. Sandberg, and H. Dehmelt. Shelved optical electron amplifier: Observation of quantum jumps. Physical Review Letters, 56:2797, 1986.Google ScholarCross Ref
- A. Nayak and A. Vishwanath. Quantum walk on the line, 2000. quant-ph/0010117.Google Scholar
- M. A. Nielsen and I. L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, 2000. Google ScholarDigital Library
- W. Paul. Electromagnetic traps for charged and neutral particles. Review of Modern Physics, 62:531--540, 1990.Google ScholarCross Ref
- C. Roos, T. Zeiger, H. Rohde, H. C. Nägerl, J. Eschner, D. Leibfried, F. Schmidt-Kaler, and R. Blatt. Quantum state engineering on an optical transition and decoherence in a paul trap. Physical Review Letters, 83:4713, 1999.Google ScholarCross Ref
- C. A. Sackett. Quantum information experiments with trapped ions: status and prospects. Quantum Information and Computation, 1(2):57--80, 2001. Google ScholarDigital Library
- T. Sauter, W. Neuhauser, R. Blatt, and P. E. Toschek. Observation of quantum jumps. Physical Review Letters, 57:1696, 1986.Google ScholarCross Ref
- P. W. Shor. Algorithms for quantum computation: Discrete logarithms and factoring. Proc. 35th Annual Symposium on Foundations of Computer Science, page 124, 1994.Google ScholarDigital Library
- S. Stenholm. The semiclassical theory of laser cooling. Review of Modern Physics, 58:699--739, 1986.Google ScholarCross Ref
- D. R. Sypher, I. M. Brereton, H. M. Wiseman, B. L. Hollis, and B. C. Travaglione. Read-only memory-based quantum computation: Experimental explorations using nuclear magnetic resonance and future prospects. Physical Review A, 66:012306, 2002.Google ScholarCross Ref
- T. Toffoli. Reversible computing. In J. W. de~Bakker and J. van Leeuwen, editors, Automata, Languages and Programming, page 632, 1980. Google Scholar
- B. C. Travaglione. Smale Scale Quantum Algorithms. PhD thesis, Department of Physics, University of Queensland, Queensland, Australia, August 2002.Google Scholar
- B. C. Travaglione and G. J. Milburn. Implementing the quantum random walk. Physical Review A, 65:032310, 2002.Google ScholarCross Ref
- B. C. Travaglione, M. A. Nielsen, H. M. Wiseman, and A. Ambainis. ROM-based computation: Quantum versus classical. Quantum Information and Computation, 2(4):324--332, 2002. Google ScholarDigital Library
- D. J. Wineland and W. M. Itano. Laser cooling of atoms. Physical Review A, 20:1521, 1979.Google ScholarCross Ref
Index Terms
- Designing and implementing small quantum circuits and algorithms
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