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A quantization of box-ball systems

Rei Inoue, Atsuo Kuniba, Masato Okado

Published Jan 1, 2004• 4 citations
OpenAlex

Abstract

Abstract. An L operator is presented related to an infinite dimensional limit of the fusion R matrices for Uq(A (1) n−1) and Uq(D(1) n). It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated. 1.

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In this paper we present a quantization of cellular automata. Our formalism is based on a lattice of qudits and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains, and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.

arXiv:0709.0006•Physical Review A•Sep 20, 2007•45 citations
Abstract. An L operator is presented related to an infinite dimensional limit of the fusion R matrices for Uq(A (1) n−1) and Uq(D(1) n). It is factorized into the local propagation operators which quantize the deterministic dynamics of particles and antiparticles in the soliton cellular automata known as the box-ball systems and their generalizations. Some properties of the dynamical amplitudes are also investigated. 1.
Abstract Box-ball system (BBS) is a prominent example of integrable cellular automata in one dimension connected to quantum groups, Bethe ansatz, ultradiscretization, tropical geometry and so forth. In this paper we study the generalized Gibbs ensemble of BBS soliton gas by thermodynamic Bethe ansatz and generalized hydrodynamics. The results include the solution to the speed equation for solitons, an intriguing connection of the effective speed with the period matrix of the tropical Riemann theta function, an explicit description of the density plateaux that emerge from domain wall initial conditions including their diffusive corrections.
Generalized hydrodynamics in complete box-ball system for Uq(sl^n)U_q(\widehat{sl}_n)Uq​(sln​)
We introduce the complete box-ball system (cBBS), which is an integrable cellular automaton on 1D lattice associated with the quantum group U_q(\widehat{sl}_n) U q ( s l ̂ n ) . Compared with the conventional (n-1) ( n − 1 ) -color BBS, it enjoys a remarkable simplification that scattering of solitons is totally diagonal. We also submit the cBBS to randomized initial conditions and study its non-equilibrium behavior by thermodynamic Bethe ansatz and generalized hydrodynamics. Excellent agreement is demonstrated between theoretical predictions and numerical simulation on the density plateaux generated from domain wall initial conditions including their diffusive broadening.
A method of quantization of classical soliton cellular automata (QSCA) is put forward that provides a description of their time evolution operator by means of quantum circuits that involve quantum gates from which the associated Hamiltonian describing a quantum chain model is constructed. The intrinsic parallelism of QSCA, a phenomenon first known from quantum computers, is also emphasized. Introduction. Soliton cellular automata (SCA), is a class of cellular automata[1] operating on binary sequences with an updating rule function f for each cell, that depends on past and present time cells, the number of which determines the radius r of the automaton[2]. Contrary to usual CA ( that evolve their cells by means of past time cells only), SCA exhibit a variety of evolution patterns[4, 5, 6] that is mainly known to characterize the temporal behavior of
Simple algebraic cellular automata and a diffusion confinement are used to model quantum behavior of a relativistic particle in a 1-D box. Comparisons and contrasts are drawn between such a quantum cellular automaton (QCA) and both orthodox and unorthodox interpretations of quantum mechanics. The persistent tension between deterministic pictures and probabilistic descriptions of quantum reality are clarified for the case of a 1-D quantum particle viewed as a cellular automaton surging for survival in space and time.
In this paper we present a quantization of cellular automata. Our formalism is based on a lattice of qudits and an update rule consisting of local unitary operators that commute with their own lattice translations. One purpose of this model is to act as a theoretical model of quantum computation, similar to the quantum circuit model. It is also shown to be an appropriate abstraction for space-homogeneous quantum phenomena, such as quantum lattice gases, spin chains, and others. Some results that show the benefits of basing the model on local unitary operators are shown: universality, strong connections to the circuit model, simple implementation on quantum hardware, and a wealth of applications.