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arXiv:quant-ph/0208149

A semi-quantum version of the game of Life

Adrian P. Flitney, Derek Abbott

Published Aug 26, 2002• 0 citations
arXivHTMLPDF

Abstract

A version of John Conway's game of Life is presented where the normal binary values of the cells are replaced by oscillators which can represent a superposition of states. The original game of Life is reproduced in the classical limit, but in general additional properties not seen in the original game are present that display some of the effects of a quantum mechanical Life. In particular, interference effects are seen.

DOI
10.48550/arxiv.quant-ph/0208149

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A version of John Conway's game of Life is presented where the normal binary values of the cells are replaced by oscillators which can represent a superposition of states. The original game of Life is reproduced in the classical limit, but in general additional properties not seen in the original game are present that display some of the effects of a quantum mechanical Life. In particular, interference effects are seen.
Conway's classic game of life is a two-dimensional cellular automaton in which each cell, either alive or dead, evolves according to rules based on its local environment. The semi-quantum game of life (SQGOL) is an adaptation in which each cell is in a superposed state of both dead and alive and evolves according to modified rules. Computer simulation of the SQGOL reveals remarkable complexity and previously unseen game behaviors. Systems evolve to a "quantum cloud" with a liveness distribution of mean =0.3480±0.0001=0.3480\pm0.0001=0.3480±0.0001 and standard deviation σ=0.0071σ=0.0071σ=0.0071 which is dependent solely on the evolutionary rules. Transient lifeforms emerge from the cloud. Semi-quantum still-lifes are discovered including the qutub which contains 4 live cells and 4 semi-quantum cells. A solitary qutub placed in an otherwise empty universe may act as a seed to reproduce child qutubs, one or more classical and/or semi-quantum lifeforms, oscillators, a quantum cloud or death depending on the initial state. Evolution to the quantum cloud occurs chaotically with floating-point errors providing the butterfly effect. Evolutionary outcomes scale in a fractal-like manner.
The classical "game of life" (GOL) due to Conway is a famous mathematical game constructed as a two-dimensional cellular automaton in which each cell is either alive or dead. A set of evolutionary rules determines whether a cell dies, survives or is born at each generation based on its local environment. The game of life is interesting because complexity emerges from simple rules and mimics real life in that a cell flourishes only if the environment is "just right" producing a breadth of life-like behavior including multi-cellular lifeforms. Results are presented from a quantum adaptation of the GOL which assigns a qubit to each cell which then evolves according to modified evolutionary rules. Computer simulation reveals remarkable evolutionary complexity that is distinct from the classical GOL and which mimics aspects of quantum biological processes and holds promise for the realistic simulation of species population dynamics. Liveness emerges as a probability density with universal statistical properties dependent solely on the evolutionary rules. New species of quantum lifeform are found. One quantum lifeform is shown to act as a seed to produce children, one or more classical and/or quantum lifeforms, classical oscillators, a liveness probability density or death with outcomes highly sensitive to the initial state. We observe the emergence of chaos and scaling phenomena making the quantum game of life an exciting prospect for further exploration as a model for life-like behaviors.
Classical stochastic Conway Game of Life is expressed by the dissipative Schrödinger equation and dissipative tight-binding model. This is conducted at the prize of usage of time dependent anomalous non-Hermitian Hamiltonians as with occurrence of complex value potential that do not preserve the normalization of wave-function and thus allows for mimicking creationism or annihilationism of cellular automaton. Simply saying time-dependent complex value eigenenergies are similar to complex values of resonant frequencies in electromagnetic resonant cavities reflecting presence of dissipation that reflects energy leaving the system or being pumped into the system. At the same time various aspects of thermodynamics were observed in cellular automata that can be later reformulated by quantum mechanical pictures. The usage of Shannon entropy and mass equivalence to energy points definition of cellular automata temperature. Contrary to intuitive statement the system dynamical equilibrium is always reflected by negative temperatures. Diffusion of mass, energy and temperature as well as phase of proposed wave function is reported and can be directly linked with second thermodynamics law approximately valid for the system, where neither mass nor energy is conserved. The concept of complex-valued mass mimics wave-function behavior. Equivalence an anomalous second Fick law and dissipative Schrödinger equation is given. Dissipative Conway Game of Life tight-binding Hamiltonian is given using phenomenological justification.