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From quantum cellular automata to quantum lattice gases

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Abstract

A natural architecture for nanoscale quantum computation is that of a quantum cellular automaton. Motivated by this observation, we begin an investigation of exactly unitary cellular automata. After proving that there can be no nontrivial, homogeneous, local, unitary, scalar cellular automaton in one dimension, we weaken the homogeneity condition and show that there are nontrivial, exactly unitary, partitioning cellular automata. We find a one-parameter family of evolution rules which are best interpreted as those for a one-particle quantum automaton. This model is naturally reformulated as a two component cellular automaton which we demonstrate to limit to the Dirac equation. We describe two generalizations of this automaton, the second, of which, to multiple interacting particles, is the correct definition of a quantum lattice gas.

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References

  1. P. Benioff, The computer as a physical system: A microscopic quantum mechanical Hamiltonian model of computers as represented by Turing machines,J. Stat. Phys. 22:563–591 (1980); R. Landauer, Uncertainty principle and minimal energy dissipation in the computer,Int. J. Theor. Phys. 21:283–297 (1982); R. P. Feynman, Quantum mechnical computers,Found. Phys. 16:507–531 (1986); and references therein.

    Google Scholar 

  2. R. P. Feynman, Simulating physics with computers,Int. J. Theor. Phys. 21:467–488 (1982).

    Google Scholar 

  3. D. Deutsch, Quantum theory, the Church-Turing principle and the universal quantum computer,Proc. R. Soc. Lond. A 400:97–117 (1985).

    Google Scholar 

  4. D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation,Proc. R. Soc. Lond. A 439:553–558 (1992); A. Berthiaume and G. Brassard, The quantum challenge to structural complexity theory, inProceedings of the 7th Structure in Complexity Theory Conference, (IEEE Computer Society Press, Los Alamitos, California, 1992), pp. 132–137; E. Bernstein and U. Vazirani, Quantum complexity theory, inProceedings of the 25th ACM Symposium on Theory of Computing (ACM Press, New York, 1993), pp. 11–20; D. R. Simon, On the power of quantum computation, inProceedings of the 35th Symposium on Foundations of Computer Science, S. Goldwasser, ed. (IEEE Computer Society Press, Los Alamitos, California, 1994), pp. 116–123.

    Google Scholar 

  5. P. W. Shor, Algorithms for quantum computation: Discrete logarithms and factoring, inProceedings of the 35th Symposium on Foundations of Computer Science, S. Goldwasser, ed. (IEEE Computer Society Press, Los Alamitos, California, 1994), pp. 124–134.

    Google Scholar 

  6. R. L. Rivest, A. Shamir, and L. Adleman, A method of obtaining digital signatures and public-key cryptosystems,Commun. ACM 21:120–126 (1978).

    Google Scholar 

  7. D. P. DiVincenzo, Two-bit gates are universal for quantum computation,Phys. Rev. A 51:1015–1022 (1995); J. I. Cirac and P. Zoller, Quantum computations with cold trapped ions,Phys. Rev. Lett. 74:4091–4094 (1995); A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator, J. Smolin and H. Weinfurter, Elementary gates for quantum computation,Phys. Rev. A 52:3457–3467 (1995); I. L. Chuang and Y. Yamamoto, A simple quantum computer,Phys. Rev. A,52:3489–3496 (1995).

    Google Scholar 

  8. W. G. Teich, K. Obermeyer, and G. Mahler, Structural basis of multistationary quantum systems. II. Effective few-particle dynamics,Phys. Rev. B 37:8111–8121 (1988).

    Google Scholar 

  9. C. S. Lent and P. D. Tougaw, Logical devices implemented using quantum cellular automata,J. Appl. Phys. 75:1818–1825 (1994).

    Google Scholar 

  10. W. G. Teich and G. Mahler, Stochastic dynamics of individual quantum systems: Stationary rate equations,Phys. Rev. A 45:3300–3318 (1992) H. Körner and G. Mahler, Optically driven quantum networks: Applications in molecular electronics,Phys. Rev. B 48:2335–2346 (1993).

    Google Scholar 

  11. W. D. Hillis, New computer architectures and their relationship to physics or why computer science is no good,Int. J. Theor. Phys. 21:255–262 (1982); N. Margolus, Parallel quantum computation, inComplexity, Entropy, and the Physics of Information, W. H. Zurek, ed. (Addison-Wesley, Redwood City, California, 1990), pp. 273–287; B. Hasslacher, Parallel billiards and monster systems, inA New Era in Computation, N. Metropolis and G.-C. Rota, eds. (MIT Press, Cambridge, Massachussetts, 1993), pp. 53–65; M. Biafore, Cellular automata for nanometer-scale computation,Physica D 70:415–433 (1994); R. Mainieri, Design constraints for nanometer scale quantum computers, preprint LA-UR 93-4333, [cond-mat/9410109] (1993).

    Google Scholar 

  12. S. Ulam, Random processes and transformations inProceedings of the International Congress of Mathematicians, L. M. Graves, E. Hille, P. A. Smith and O. Zariski, eds. (AMS, Providence, Rhode Island, 1952), Vol II, pp. 264–275. J. von Neumann,Theory of Self-Reproducing Automata, edited and completed by A. W. Burks (University of Illinois Press, Urbana, Illinois, 1966).

    Google Scholar 

  13. G. Grössing and A. Zeilinger, Quantum cellular automata,Complex Systems 2:197–208 (1988).

    Google Scholar 

  14. S. Fussy, G. Grössing, H. Schwabl and A. Scrinzi, Nonlocal computation in quantum cellular automata,Phys. Rev. A,48:3470–3477 (1993).

    Google Scholar 

  15. K. Morita and M. Harao, Computation universality of one-dimensional reversible (injective) cellular automata,Trans. IEICE Japan E 72:758–762 (1989).

    Google Scholar 

  16. G. V. Riazanov, The Feynman path integral for the Dirac equation,Sov. Phys. JETP 6:1107–1113 (1958); R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965), pp. 34–36.

    Google Scholar 

  17. J. Hardy, Y. Pomeau, and O. de Pazzis, Time evolution of a two-dimensional model system. I. Invariant states and time correlation functions,J. Math. Phys. 14:1746–1759. (1973); J. Hardy, O. de Pazzis, and Y. Pomeau, Molecular dynamics of a classical lattice gas: Transport properties and time correlation functions,Phys. Rev. A 13:1949–1961 (1976); U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier-Stokes equation,Phys. Rev. Lett. 56:1505–1508 (1986).

    Google Scholar 

  18. S. Succi and R. Benzi, Lattice Boltzmann equation for quantum mechanics,Physica D 69:327–332 (1993); S. Succi, Numerical solution of the Schroedinger equation using a quantum lattice Boltzmann equation, preprint [comp-gas/9307001] (1993).

    Google Scholar 

  19. I. Bialynicki-Birula, Weyl, Dirac, and Maxwell equations on a lattice as unitary cellular automata,Phys. Rev. D 49:6920–6927 (1994).

    Google Scholar 

  20. R. Landauer, Is quantum mechanics useful?Phil. Trans. R. Soc. Lond. A,353:367–376 (1995).

    Google Scholar 

  21. M. B. Plenio and P. L. Knight, Realistic lower, bounds for the factorization time of large numbers on a quantum computer, preprint [quant-ph/9512001] (1995); D. Beckman, A. N. Chari, S. Devabhaktuni, and J. Preskill, Efficient networks for quantum factoring, preprint CALT-68-2021, [quant-ph/9602016] (1996).

  22. W. G. Unruh, Maintaining coherence, in quantum computers,Phys. Rev. A,51:992–997, (1995); G. M. Palma, K.-A. Souminen and A. Ekert, Quantum computers and dissipation,Proc. R. Soc. Lond. A 452:567–584 (1996).

    Google Scholar 

  23. I. L. Chuang, R. Laflamme, P. Shor, and W. H. Zurek, Quantum computers, factoring and decoherence,Science 270:1633–1635 (1995); C. Miquel, J. P. Paz, and R. Perazzo, Factoring in a dissipative quantum computer, preprint [quant-ph/9601021] (1996).

    Google Scholar 

  24. H. Weyl,The Theory of Groups and Quantum Mechanics (Dover, New York, 1950).

    Google Scholar 

  25. S. Wolfram, Computation theory of cellular automata,Commun. Math. Phys. 96:15–57 (1984).

    Google Scholar 

  26. P. Ruján, Cellular automata and statistical mechanical models,J. Stat. Phys. 49:139–222. (1987); A. Georges and P. Le Doussal, From equilibrium spin models to probabilistic cellular automata,J. Stat. Phys. 54:1011–1064 (1989).

    Google Scholar 

  27. T. Toffoli and N. H. Margolus, Invertible cellular automata: A review,Physica D 45:229–253 (1990).

    Google Scholar 

  28. Y. L. Luke,The Special Functions and Their Approximations (Academic Press, New York, 1969), Vol. I, p. 49.

    Google Scholar 

  29. T. Jacobson and L. S. Schulman, Quantum stochastics: the passage from a relativistic to a non-relativistic path integral,J. Phys. A: Math. Gen. 17:375–383 (1984).

    Google Scholar 

  30. D. A. Meyer, In preparation.

  31. B. Hasslacher and D. A. Meyer, Lattice gases and exactly solvable models,J. Stat. Phys. 68:575–590 (1992).

    Google Scholar 

  32. R. J. Baxter,Exactly Solved Models in Statistical Mechanics (Academic Press, New York, 1982).

    Google Scholar 

  33. C. Destri and H. J. de Vega, Light-cone lattice approach to fermionic theories in 2D,Nucl. Phys. B 290:363–391 (1987).

    Google Scholar 

  34. D. Kandel, E. Domany, and B. Nienhuis, A six-vertex model as a diffusion problem: Derivation of correlation functions,J. Phys. A: Math. Gen. 23:L755-L762 (1990); P. Orland Six-vertex models as Fermi gases,Int. J. Mod. Phys. B 5:2385–2400 (1991).

    Google Scholar 

  35. M. Hénon, On the relation between lattice gases and cellular automata; inDiscrete Kinetic Theory, Lattice Gas Dynamics and Foundations of Hydrodynamics, R. Monaco, ed. (World Scientific, Singapore, 1989), pp. 160–161.

    Google Scholar 

  36. H. Hrgovčić, Quantum mechanics on a space-time lattice using path integrals in a Minkowski metric,Int. J. Theor. Phys. 33:745–795 (1994); T. M. Samols, A stochastic model of a quantum field theory,J. Stat. Phys. 80:793–809 (1995).

    Google Scholar 

  37. H. B. Nielsen and M. Ninomiya, A no-go theorem for regularizing chiral fermions,Phys. Lett. B 105:219–223 (1981), and references therein.

    Google Scholar 

  38. Y. Nakawaki, A new choice for two-dimensional Dirac equation on a spatial lattice,Prog. Theor. Phys. 61:1855–1857 (1979); R. Stacey, Eliminating lattice fermion doubling,Phys. Rev. D 26:468–472 (1982); J. M. Rabin, Homology theory of lattice fermion doubling,Nucl. Phys. B 201:315–332 (1982).

    Google Scholar 

  39. L. Susskind, Lattice fermions,Phys. Rev. D. 16:3031–3039 (1977).

    Google Scholar 

  40. K. G. Wilson, Confinement of quarks,Phys. Rev. D 10:2445–2459 (1974).

    Google Scholar 

  41. L. Bombelli, J. Lee, D. A. Meyer, and R. D. Sorkin Spacetime as a causal set,Phys. Rev. Lett. 59:521–524 (1987); D. A. Meyer, Spacetime Ising models, UCSD preprint (1995); D. A. Meyer, Induced actions for causal sets, UCSD preprint (1995).

    Google Scholar 

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  1. Project in Geometry and Physics, Department of Mathematics, University of California-San Diego, 92093-0112, La Jolla, California

    David A. Meyer

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Meyer, D.A. From quantum cellular automata to quantum lattice gases. J Stat Phys 85, 551–574 (1996). https://doi.org/10.1007/BF02199356

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