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Quantum Game of Life

2012; Institute of Physics; Volume: 97; Issue: 2 Linguagem: Inglês

10.1209/0295-5075/97/20012

1286-4854

D. Bleh, Tommaso Calarco, Simone Montangero,

Cellular Automata and Applications

W e introduce aq uantum version of the Game of Life and we use it to study the emergence of complexity in a quantum world. We show that the quantum evolution displays signatures of complex behaviour similar to the classical one, however a regime exists, where the quantum Game of Life creates more complexity, in terms of diversity, with respect to the corresponding classical reversible one. Copyright c EPLA, 2012 The Game of Life (GoL) has been proposed by Conway in 1970 as a wonderful mathematical game which can describe the appearance of complexity and the evolution of under some simple rules (1). Since its introduction it has attracted a lot of attention, as despite its simplicity, it can reveal complex patterns with unpredictable evolu- tion: From the very beginning a lot of structures have been identified, from simple blinking patterns to complex evolving figures which have been named blinkers, glid- ers up to spaceships due to their appearance and/or dynamics (2). The classical GoL has been the subject of many studies: It has been shown that cellular automata defined by the GoL have the power of a Universal Turing machine, that is, anything that can be computed algo- rithmically can be computed within Conway's GoL (3,4). Statistical analysis and analytical descriptions of the GoL have been performed; many generalisations or modifica- tions of the initial game have been introduced as, for exam- ple, a simplified one-dimensional version of the GoL and a semi-quantum version (5-7). Finally, to allow a statistical- mechanics description of the GoL, stochastic components have been added (8). In this letter, we bridge the field of complex systems with quantum mechanics introducing a purely quantum GoL and we investigate its dynamical properties. We show that it displays interesting features in common with its classical counterpart, in particular regarding the variety of supported dynamics and different behaviour. The system converges to a quasi-stationary configuration in terms of macroscopic variables, and these stable configurations depend on the initial state, e.g., the initial density of sites for random initial configurations. We show that simple, local rules support complex behaviour and that the diversity of the structures formed in the steady state resembles that of the classical GoL, however a regime exists where quantum dynamics allows more diversity to be created than possibly reached by the classical one. The universe of the original GoL is an infinite two-dimensional orthogonal grid of square cells with coordination number eight, each of them in one of two possible states, alive or (1). At each step in time, the pattern present on the grid evolves instantaneously following simple rules: any cell with exactly three live neighbours comes to life; any live cell with less than two or more than three live neighbours dies as if by loneliness or overcrowding. As already pointed out in (7), the rules of the GoL are irreversible, thus their generalisation to the quantum case implies rephrasing them to make them compatible with a quantum reversible evolution. The system under study is a collection of two-level quantum systems, with two possible orthogonal states, namely the state dead (|0� ) and (|1� ). Clearly, differently from the classical case, a site can be also in a superpo- sition of the two possible classical states. The dynamics is defined as follows in terms of the GoL language: a site with two or three neighbouring alive sites is active, where active means that it will come to life and eventually die on a typical timescale T (setting the problem timescale, or time between subsequent generations). That is, if maintained active by the surrounding conditions, the site will complete a full rotation, if not, it is frozen in its state. Stretching the analogy with Conway's GoL to the limit, we are describing the evolution of a virus culture: each individual undergoes its life cycle if the environment allows it, otherwise it hibernates in its current state and waits for conditions to change such that the site may