Quantum emulation of classical dynamics

In statistical mechanics, it is well known that finite-state classical lattice models can be recast as quantum models, with distinct classical configurations identified with orthogonal basis states. This mapping makes classical statistical mechanics on a lattice a special case of quantum statistical mechanics, and classical combinatorial entropy a special case of quantum entropy. In a similar manner, finite-state classical dynamics can be recast as finite-energy quantum dynamics. This mapping translates continuous quantities, concepts and machinery of quantum mechanics into a simplified finite-state context in which they have a purely classical and combinatorial interpretation. For example, in this mapping quantum average energy becomes the classical update rate. Interpolation theory and communication theory help explain the truce achieved here between perfect classical determinism and quantum uncertainty, and between discrete and continuous dynamics.

[1]  F. Stenger.  Numerical Methods Based on Whittaker Cardinal, or Sinc Functions , 1981 .

[2]  Norman Margolus,et al.  Universal Cellular Automata Based on the Collisions of Soft Spheres , 2008, Collision-Based Computing.

[3]  Protopopescu.  (Discrete kinetic theory, lattice gas dynamics and foundations of hydrodynamics) , 1988 .

[4]  N. Margolus.  Parallel Quantum Computation ∗ , 1990 .

[5]  L. Kadanoff,et al.  Reversible Boolean networks I: distribution of cycle lengths Physica D 149 , 2000, cond-mat/0004422.

[6]  David P. DiVincenzo,et al.  Quantum information and computation , 2000, Nature.

[7]  M. Creutz.  Deterministic Ising dynamics , 1986 .

[8]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[9]  N. Margolus.  Mechanical Systems that are both Classical and Quantum , 2008, 0805.3357.

[10]  W. L. Cowley.  The Uncertainty Principle , 1949, Nature.

[11]  John Ziman,et al.  Elements Of Advanced Quantum Theory , 1969 .

[12]  Gerard 't Hooft,et al.  Equivalence relations between deterministic and quantum mechanical systems , 1988 .

[13]  U. Frisch,et al.  Lattice gas models for 3D hydrodynamics , 1986 .

[14]  Stephen M. Barnett,et al.  Quantum Optical Phase , 1997 .

[15]  Seth Lloyd,et al.  Quantum theory of optical temporal phase and instantaneous frequency , 2008 .

[16]  T. Toffoli,et al.  Conservative logic , 2002, Collision-Based Computing.

[17]  J. Hilgevoord,et al.  Uncertainty principle and uncertainty relations , 1985 .

[18]  Norman Margolus,et al.  First-moment distinguishability bounds , 2011, ArXiv.

[19]  P. Benioff.  Quantum Mechanical Models of Turing Machines That Dissipate No Energy , 1982 .

[20]  Norman H. Margolus,et al.  Crystalline computation , 1998, comp-gas/9811002.

[21]  F. Khalili,et al.  Quantum nondemolition measurements: the route from toys to tools , 1996 .

[22]  Jeffrey Yepez,et al.  Integer lattice gases , 1997 .

[23]  G. Vahala,et al.  Superfluid turbulence from quantum Kelvin wave to classical Kolmogorov cascades. , 2009, Physical review letters.