D. A. Meyer. From quantum cellular automata to quantum lattice gasses. J. Stat. Phys., 85:551--574, 1996.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....It would be interesting to see how much of this carries over to the quantum domain. There are many proposals for computation, both classical and quantum, using quantum 63 cellular automata (see, e.g. Briegel and Raussendorf [18] Fussy et al. 44] Lent et al. 77] Lloyd [84] or Meyer [88]) The two main attractions of quantum cellular automata are (a) massively parallel structure, and (b) the possibility of a phase transition. We have already discussed massively parallel structure of quantum cellular automata in Section 3.4.3; here we focus our attention on phase transitions. A ....

D.A. Meyer, \From quantum cellular automata to quantum lattice gases," J. Stat. Phys. 85, 551 (1996).

....impossible, since the global process is non unitary. It is also easy to verify that the only possible homogeneous (i.e. translationally invariant) unitary processes on the line involving transitions between adjacent lattice sites are the left and right shift operators (up to an overall phase) [13]. This corresponds to the rather uninteresting motion in a single direction. As explained below (and also shown by [13] it is still possible to construct a unitary walk if the particle has an extra degree of freedom that assists in its motion. Consider a quantum particle that moves freely on ....

....translationally invariant) unitary processes on the line involving transitions between adjacent lattice sites are the left and right shift operators (up to an overall phase) 13] This corresponds to the rather uninteresting motion in a single direction. As explained below (and also shown by [13]) it is still possible to construct a unitary walk if the particle has an extra degree of freedom that assists in its motion. Consider a quantum particle that moves freely on the integer points on the line, and has an additional degree of freedom, its chirality, which takes values left and ....

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David A. Meyer. From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics, 85:551--574, 1996.

....impossible, since the global process is necessarily non unitary. It is also easy to verify that the only possible translationally invariant unitary processes on the line allowing only transitions between adjacent lattice sites are the left and right shift operators, up to an overall phase [24]. These processes simply correspond to motion in a single direction. If the particle has an extra degree of freedom that assists in its motion, however, then it is possible to construct more interesting translation invariant local unitary processes. Consider a quantum particle that moves freely ....

....[4, 5] This gives another asymptotic form for the probability distribution. The Schr odinger approach is also quite general and could be potentially applied to quantum walks on any Cayley graph. Related work Various quantum variants of random walks have previously been studied by a few authors [6, 12, 24, 32], but their results are, for the most part, unrelated to ours. The rst study of quantum walks is apparently due to Meyer [24] Meyer s model (quantum lattice gas automata or QLGA) is equivalent to our two way in nite Hadamard walk, but he addresses di erent questions than the ones we consider. ....

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D. Meyer. From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics, 85:551-574, 1996. Also available from the Los Alamos Preprint Archive, quant-ph/9604003.

....it should be noted that we are not suggesting that this is the only de nition there are several ways one can de ne quantum variations on random walks (see, for instance, the quantum processes considered by Farhi and Gutmann [4] 1. 1 Related work Quantum random walks have been studied by Meyer [11] and Farhi and Gutmann [4] but their results are mostly unrelated to ours. Meyer s model (quantum lattice gas automata or QLGA) is the same as our two way in nite quantum random walk but the questions that he asks are quite di erent from ours. The only overlapping result is the formula for the ....

....square of the absolute value of the amplitude, bounds on amplitudes imply bounds on the probabilities. To reach jx; 0i or jx; 1i in t steps, there should be m = x t 2 moves left and t m = t x 2 moves right. By counting the paths consisting of m moves left and t m moves right, one gets Lemma 6 [11] The amplitude of jx; 0i after t applications of Uwalk is 1 p 2 t X k m 1 k t m k ( 1) m k : 1) The amplitude of jx; 1i after t applications of Uwalk is 1 p 2 t X k m 1 k 1 t m k ( 1) m k 1 : 2) This allows to calculate the amplitudes of jx; 0i and jx; ....

D. Meyer. From quantum cellular automata to quantum lattice gases. Journal of Statistical Physics, 85:551-574, 1996. Also available from the Los Alamos Preprint Archive, quantph /9604003.

.... Boghosian [Bog98] Proposed QC simulations of quantum mechanical systems include: many body systems (Wiesner[Wie96] many body Fermi systems (Abrams, Lloyd [AL97] multiparticle (ballistic) evolution (Benioff [Ben96] quantum lattice gas models (Boghosian, Taylor [BT96] Meyer [Mey96a, Mey96b] Ising spin glasses (Lidar, Biham [LB97] the thermal rate constant (Lidar, Wang [LW98] quantum chaos (Schack [Sch97] ffl Quantum Cryptography. Bennett et al. [BBB 82] gave the first methods for quantum cryptography using qubits as keys, which are secure against all possible types of ....

D. A. Meyer, From quantum cellular automata to quantum lattice gases, (Online preprint quant-ph/9604003), J. Stat. Phys. 85, (1996) 551-574.

....theories) In this paper we describe in some detail a class of algorithms for simulating the many body Schrodinger equation. These algorithms were presented from a different point of view in [5,6] Based on simple Quantum Cellular Automata (QCA) 7] and Quantum Lattice Gas Automata (QLGA) [8] models, these algorithms can be used to simulate systems of interacting nonrelativistic quantum particles with speedup exponential in the number of particles in the system. We give a short discussion of progress which has been made toward developing algorithms for simulating systems of many Dirac ....

.... Delta 0 a b a a 0 0 0 Delta Delta Delta 0 0 a b 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A (7) where a; b are complex numbers satisfying jbj 2 2ja 2 j = 1 and a b ab = 0. Unfortunately, this operator cannot be implemented by a simple set of two q bit operations [7,8]. One way of seeing that this is not possible is to take the inverse of the matrix M . The matrix M Gamma1 is a dense matrix for generic lattice size l, indicating that M cannot be implemented by performing some sequence of local two q bit operations at each lattice site in a way which is ....

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D. Meyer, From quantum cellular automata to quantum lattice gases, UCSD preprint quant-ph/9604003, March 1996.

....useful to bridge the gap between physics and computational theory. From this intermediate standpoint there are at least two directions possible, each with its own merits. By going from qca to physics one obtains a powerful model to describe quantum mechanical systems. Recent work by David Meyer [39, 40, 41, 42] goes along this pathway. The other direction is to try to actually construct a controllable qca. Several authors have suggested that qca like systems are more likely to be build than quantum Turing machines oriented structures [8, 34, 35, 36] If such a construction would indeed be possible in ....

David A. Meyer. From quantum cellular automata to quantum lattice gases. http://xxx. lanl.gov/abs/quant-ph/9604003, March 1996. To appear in Journal of Statistical Physics.

.... Boghosian [Bog98] Proposed QC simulations of quantum mechanical systems include: many body systems (Wiesner[Wie96] many body Fermi systems (Abrams, Lloyd [AL97] multiparticle (ballistic) evolution (Benioff [Ben96] quantum lattice gas models (Boghosian, Taylor [BT96] Meyer [Mey96a, Mey96b] Ising spin glasses (Lidar, Biham [LB97] the thermal rate constant (Lidar, Wang [LW98] quantum chaos (Schack [Sch97] ffl Quantum Learning. QC may have some interesting applications the learning theory and related problems. Bshouty, Jackson [BJ95] describe learning Boolean formulas in ....

D. A. Meyer, From quantum cellular automata to quantum lattice gases, (Online preprint quant-ph/9604003), J. Stat. Phys. 85, (1996) 551-574.

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", quant-ph/9604003; J. Statist. Phys. 85 (1996) 551--574.

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Statist. Phys. 85 (1996) 551--574.

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Statist. Phys. 85 (1996) 551--574.

....Aharonov Bohm e#ect; spatial topology. 1 Topology computation David A. Meyer Quantum lattice gas automata (QLGA) have been proposed as a possible architecture for solid state quantum computers since they require only an array of sites which can support an extended (multi )electron wave function [1]. The simplicity of such an architecture makes nanoscale fabrication plausible [2] The main incentives for pursuing the program of quantum computation, however, are the quantum algorithms of Shor [3] and Grover [4] for example, which provide substantial improvements over deterministic or ....

....translations analogous to deterministic billiard ball models for universal computation [6] seem impractical to implement in QLGA models. Just as deterministic LGA e#ciently simulate fluid flow in certain parameter regimes [7] QLGA seem best suited for simulation of quantum physical processes [1,8]. Single particle QLGA have been shown, in fact, to limit to the Schrodinger [9] and Dirac [1] equations under appropriate conditions. In this letter we show that in simulating a single quantum particle, QLGA e#ciently perform a new and interesting computation of spatial topology. We draw ....

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

....of the Dirac equation it is straightforward to compute the propagator K ff 2 ff 1 (x 2 ; t 2 ; x 1 ; t 1 ) satisfying ff 2 (x 2 ; t 2 ) Z x2 Deltat x2 Gamma Deltat K ff 2 ff 1 (x 2 ; t 2 ; x 1 ; t 1 ) ff 1 (x 1 ; t 1 ) dx 1 ; 2) where Deltat : t 2 Gamma t 1 . The result is [8,9,10]: K ff 2 ff 1 (x 2 ; t 2 ; x 1 ; t 1 ) ae ffi( Deltat Gamma ff 1 Deltax) Gamma ( Deltat ff 1 Deltax)mJ 1 (m ) 2 if ff 2 = ff 1 ; imJ 0 (m ) 2 if ff 2 6= ff 1 , 3) where Deltax : x 2 Gamma x 1 , p ( Deltat) 2 Gamma ( Deltax) 2 and J i denotes the i th order Bessel ....

....not only does the initial increase occur faster, but the entropy also reaches greater values before starting to decrease. Spacetime plots of quantum lattice gas automaton simulations of the Dirac equation showing exactly this behavior (although with periodic spatial coordinate) can be found in [10]. Decoherence in the Dirac equation David A. Meyer 0 0.5 1 1.5 2 t 0.2 0.4 0.6 0.8 1 S Figure 4. Entropy plotted as a function of time for an initial position Gaussian with m = 1. The inset graphs show the chirality position distributions at half integer times. The results shown in Figures 1 4 ....

D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

....to be the probability that a stochastic particle is in cell x at time t. If the field is complex valued, or more precisely, if S = fz 2 C j jzj 1g, and the evolution matrix is unitary then (1) defines what we refer to here as a scalar unitary CA; this is a special case of a quantum CA (QCA) [3,4,5,6]. Unitary evolution preserves the L 2 norm of OE: P x jOE(x)j 2 ) 1=2 ; if the L 2 norm of OE 0 is 1, then OE t (x) is the amplitude for the system to be in, and jOE t (x)j 2 is the probability of observing, the state x at time t. Scalar QCA were first considered by Grossing and ....

....t. Scalar QCA were first considered by Grossing and Zeilinger [4] although they found nontrivial homogeneous scalar CA in one dimension with neighborhoods of radius one (i.e. with the evolution matrix tridiagonal) only by relaxing their definition to allow approximately unitary evolution. In [3] we showed that only trivial homogeneous scalar unitary CA exist in one dimension with neighborhoods of any size: NO GO LEMMA. In one dimension there exists no nontrivial, homogeneous, scalar unitary CA. More explicitly, every band r diagonal unitary matrix which commutes with the 1 step ....

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

....we find a complete set of invariants. These conserved quantities bear interpretation as energy and momentum, and enable the exact computation of the evolution. This model has been observed to simulate both the (nonrelativistic) Schrodinger equation [6,7] and the (relativistic) Dirac equation [8,9,10]. Although perhaps surprising at first, this is not inconsistent as the Dirac equation is well known to have a nonrelativistic limit [11] and, as Benzi and Succi have emphasized, numerical evolution of parabolic equations can be stabilized and localized by the use of artificially hyperbolic ....

....evolve as U jx; ffi = ajx ff; ffi bjx ff; Gammaffi: 2:2) That is, an initially right (left) moving quantum particle moves one lattice point to the right (left) and then continues in the same direction with amplitude a or changes direction with amplitude b as shown in Figure 1. We showed in [10] that the most general unitary evolution for a one dimensional QLGA with parity invariance and particle speed 1 is described in the one particle sector by a unitary scattering matrix S : b a a b : 2:3) i.e. invariance under x Gammax; also called reflection invariance. QLGA and ....

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

....and standard formulations of quantum mechanics in the continuum. Our recent work explaining the necessity of non unitarity in earlier attempts of Grossing, Zeilinger, et al. to construct homogeneous CA for quantum evolution [5] demonstrates this equivalence directly for the discrete models [6]. We also note that, in contrast to simulation with deterministic or probabilistic LGA, simulation with a QLGA requires evolution along all possible spacetime trajectories. This may be achieved (slowly) by evolution of the quantum lattice Boltzmann equation on a classical computer or, at present ....

....effectively by QLGA, but also how well, as Feynman suggested [9] a quantum computer might simulate physics. In addition, we expect the quantum mechanics of LGA to have implications for discrete models of fundamental physics: we have already found remarkable consequences of unitarity in linear [6,10] and nonlinear [11] QCA. We begin in the next section by recalling the model of [6] with which we will be working: the most general one dimensional homogeneous QLGA with a single particle of speed no more than 1 in lattice units. The local evolution rule for this model has two free Quantum ....

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

.... local, unitary rule would have the quantum version of the massively parallel architecture possessed, for example, by Margolus CAM machines [9] The simplest algorithms which would run on such an architecture are quantum cellular automata (QCA) 10] or quantum lattice gas automata (QLGA) [11]. Even in d = 1 spatial dimensions QCA are capable of universal computation [12] and the existence of the universal reversible billiard ball computer [13] implies that QLGA are also, at least in d 2 spatial dimensions. Just as classical LGA are most effectively deployed to simulate physical ....

....of the universal reversible billiard ball computer [13] implies that QLGA are also, at least in d 2 spatial dimensions. Just as classical LGA are most effectively deployed to simulate physical systems such as fluid flow [14] however, QLGA most naturally simulate quantum physical systems [11,15,16]: with the simplest homogeneous evolution rule, one particle QLGA simulate the constant potential Dirac [11] or Schrodinger [17] equation, depending on the relative scaling of the lattice spacing and timestep. An earlier paper [15] initiated a project to analyze which physical processes QLGA can ....

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D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Stat. Phys. 85 (1996) 551--574.

....Hasslacher Meyer 1. Introduction Lattice gas automata (LGA) 1,2] have successfully modelled a variety of fluid mechanics systems: low Reynolds number flow in complicated geometry [3] multiphase flow [4] micellular assembly [5] and even, transformed into discrete quantum systems, the Dirac [6] and Schrodinger [7] equations. All of these systems, however, consist of flow in a fixed background geometry which is represented in the LGA models by a fixed lattice (and boundary conditions) In contrast, there are many natural systems with dynamical geometry. These range from the biological ....

....(difference) vectors. The evolution rule has two phases: first each particle advects to the lattice point obtained by adding its momentum to its current position; second, the particles at each lattice point scatter according to some deterministic [1] probabilistic [2] or quantum mechanical [6,7] rule. The advection phase is trivially reversible and the scattering phase will be also, provided it is described by a permutation, a doubly stochastic, or a unitary matrix, respectively. We remark, however, that physical time reversibility is not achieved exactly by (parity) inverting all ....

D. A. Meyer, "From quantum cellular automata to quantum lattice gases", J. Statist. Phys. 85 (1996) 551--574.

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D. A. Meyer. From quantum cellular automata to quantum lattice gasses. J. Stat. Phys., 85:551--574, 1996.

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D. A. Meyer. From quantum cellular automata to quantum lattice gasses. J. Stat. Phys., 85:551--574, 1996.

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Meyer, D., "From quantum cellular automata to quantum lattice gases." quantph /9604003.

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D. Meyer, "From quantum cellular automata to quantum lattice gases," UCSD preprint quant-ph/9604003, March 1996.

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