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arXiv:1902.07834

Life, death, new species and children from a "Quantum Game of Life"

David A. Faux, Mayank Shah, Christopher Knapp

Published Feb 19, 2019• 0 citations
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.1902.07834

Related papers

The semi-quantum game of life

David A. Faux, Mayank Shah, Christopher Knapp

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 $=0.3480\pm0.0001$ and standard deviation $σ=0.0071$ 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.

arXiv:1902.07835•Feb 19, 2019•0 citations

Towards quantization Conway Game of Life

Krzysztof Dominik Pomorski, Dariusz Kotula

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.

arXiv:2306.15151•Jun 27, 2023•0 citations

Quantum Artificial Life Takes Its First Step on an IBM Quantum Computer

inquantio

A research team based in Spain modeled the four ingredients of Darwinian evolution, self replication, mutation, interaction between individuals, and death, as a quantum circuit and ran it for the first time on IBM's cloud quantum computer, ibmqx4. Each individual was defined as two qubits, a genotype and a phenotype, and successive generations were linked by partial quantum cloning, a procedure that works around the no cloning theorem. The complete model combining all four ingredients reproduced the ideal probability distribution with a fidelity of 93.94%, though the researchers were explicit that this is a proof of principle, a simulation of the logic of life on quantum hardware rather than an account of how life began. [Quantum Biology Society] Where quantum biology has mostly gone looking for quantum mechanical phenomena inside living organisms, this study runs the argument in the opposite direction: it implements the logic of biological evolution inside the physical environment of a quantum computer. The authors describe their line of work as quantum biomimetics, the design of quantum algorithms that imitate biological processes. Researchers at the University of the Basque Country (UPV/EHU) in Bilbao, together with colleagues at several partner institutions, designed the elements of Darwinian natural selection and evolution (self replication, mutation, interaction, and death) as a quantum algorithm that works around the no cloning theorem, one of the founding principles of quantum mechanics. They then succeeded in running it on ibmqx4, the five qubit superconducting chip IBM makes available over the cloud, putting into practice a model the same group had proposed two years earlier. The results, published in Scientific Reports in 2018, stand as the first experimental implementation of a quantum artificial life algorithm on quantum hardware. ■ Two Qubits per Individual: Partial Quantum Cloning Sidesteps the No Cloning Theorem The smallest unit of life in this model, the individual, is built from two qubits, one holding the genotype and one the phenotype. The genotype carries the information that identifies the type of living unit and is handed on from generation to generation. The phenotype is set by the genetic information together with the interaction between the individual and its environment, and that state, along with the information it encodes, is degraded over the individual's lifetime, which is how aging and death enter the model. Self replication, the core of life, is implemented through partial quantum cloning. Quantum mechanics forbids making a perfect copy of an unknown quantum state in full, but it does permit the expectation value of a single chosen observable, here the Pauli Z observable σz, to be written into a blank qubit. Using controlled NOT (CNOT) gates, the researchers passed that particular piece of the genotype's information first into the genotype qubit of the next generation and then into its phenotype qubit, completing a system of heredity in the quantum world. ■ A Practical Compromise with the Hardware: Emulating Dissipation and Mutation Because the flexibility of current quantum hardware is limited, death and mutation had to be simulated by practical approximations. Death should be produced by leaving the phenotype qubit open to its environment so that its information gradually dissipates, but the device could not perform the conditional projection of the phenotype onto the |0⟩ state that this calls for. Instead the researchers applied a sequence of small rotations about the σy axis, one for each simulated time step, to imitate the running out of a lifetime. As they note, this replaces the dissipation with a discrete process and loses its exponential character. Mutation, likewise, was not implemented as a fully random rotation. The researchers fixed the mutation to a single bit flip (σx) and then took a statistical detour: circuits with and without a mutation were run different numbers of times and the data merged afterward so that the intended mutation rate came out of the aggregate. Interaction between individuals is a conditional operation that swaps phenotypes according to the genotypes, and it was built by combining two qubit gates including CNOT, SWAP, and the controlled square root of NOT. ■ The Results, and the Mutation Paradox The experiments proceeded by adding the elements of evolution one after another: interaction between individuals, self replication, dissipation into the environment, and mutation. For the complete model with all four in place, the measured probability distribution matched the ideal prediction with a fidelity of 93.94%. One result along the way was especially curious. When mutation, the element that injects randomness into the system, was included, the fidelity of the data went up rather than down. The researchers give a level headed account of it: mutations tend to homogenize the probability distribution, which is the same effect the errors already present in the experimental platform produce. They also remark on the intriguing parallel with the classical world, where mutation likewise helps a species adapt to changing environments. ■ What Is Actually Quantum Here: Entanglement That Remembers a Lineage The simpler statistical indicators can be reproduced by a classical computer model without much difficulty. Every indicator based on measurements in the σz basis, the researchers point out, can be matched by a classical probability distribution. So which part of this model is genuinely quantum? The answer shows up in the correlations. A partial quantum cloning operation entangles the qubits it involves, carrying the expectation value of σx in the original state into the joint value of σx⊗σx. In a simplified case without mutations or interactions between individuals, measuring σx⊗σx⊗σx⊗σx across the four qubits of two individuals gives α if the two are linked by a self replication event, and α² if they are not causally connected, where α is the expectation value of σx in the precursor genotype. The two histories are therefore distinguishable from the correlation alone. Had the precursor genotype been a classical incoherent mixture instead, that expectation value would be zero and nothing would propagate into these global correlations at all. In other words, without keeping a separate ledger of who descended from whom, as a classical model would have to, the entangled state of the individuals itself records and attests to the lineage. On the chip this signature survived only in part: the measured value of σx⊗σx⊗σx⊗σx was 0.22 against an ideal 0.56, a sizable error. It was nonetheless clearly different from the product of the correlations within each individual taken separately, which comes out at zero, and it is that difference which shows the two individuals to be causally related. ■ Significance and Limits The study is a well executed proof of principle that a microscopic quantum system can hold the grammar of Darwinian evolution consistently, and that it can be run on a real superconducting quantum chip. The paper is explicit, however, that it is not an observation of quantum effects at work in a living organism. The authors leave open the philosophical question of whether the origin of life is genuinely quantum mechanical, and they state that the quantum supremacy referred to in the abstract and conclusion was not achieved in this experiment but remains a target to be explored on larger quantum computers in the future. They also acknowledge that the part of the dynamics carried by σz, which encodes the lifetime and the predator prey character of an individual, is classical information that a simpler classical analogue could predict; what the classical version cannot supply is the entanglement linking the living units. In asking whether the logic of life can be recoded in the language of quantum mechanics, this work sets out another blueprint for where quantum algorithms and the biological sciences may converge. #QuantumBiology #QuantumComputer #IBMQuantumComputer #QuantumArtificialLife #ArtificialLife #Qubit #QuantumEntanglement #NoCloningTheorem #DarwinianEvolution https://www.nature.com/articles/s41598-018-33125-3

•Jul 21, 2026•0 citations

A semi-quantum version of the game of Life

Adrian P. Flitney, Derek Abbott

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.

arXiv:quant-ph/0208149•Aug 26, 2002•0 citations

A Semi-quantum Version of the Game of Life

Adrian P. Flitney, Derek Abbott

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.

Birkhäuser Boston eBooks•Dec 23, 2007•4 citations

Hybrid Quantum Games

Shuliang Li, D. Huang

By incorporating payoff matrix and population evolution of open systems and proposing density games, pioneering effort has been made to break the following two basic assumptions of evolutionary game theory over the past decades: the payoff from the game determines the reproductive rate of individuals, and the total population size remains constant by a variable called the death rate. On the other hand, most studies on strategic interactions in game theories consider independent individual strategies as the basic unit of analyses. This is in fact not always true in human interaction strategies. It has been unclear that how subjective non-independent species with strategic interactions in open dynamic systems evolve. In this paper, we create the new concept of hybrid quantum games, and propose an open quantum dynamic symbiosis model for exploring the evolution of this type of species under unclosed social settings. We show that the stability conditions of the model with non-independent strategic interactions are influenced by the specific symbiosis states, game payoff matrix and quantum entanglement. The stability conditions of the competition scenario are more stringent than those of the cooperation one. Our system dynamics simulation study demonstrates and verifies the dynamic changes for the population sizes of two different species in the theoretical model, on the basis of the stability conditions of the competition and cooperation scenarios. Our model combines quantum games, game payoff matrix and symbiosis equations organically, and can provide insights into competition, cooperation and symbiosis mechanisms in open systems with subjective non-independent strategies considered.

•Jan 1, 2017•0 citations
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.
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.
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.
A research team based in Spain modeled the four ingredients of Darwinian evolution, self replication, mutation, interaction between individuals, and death, as a quantum circuit and ran it for the first time on IBM's cloud quantum computer, ibmqx4. Each individual was defined as two qubits, a genotype and a phenotype, and successive generations were linked by partial quantum cloning, a procedure that works around the no cloning theorem. The complete model combining all four ingredients reproduced the ideal probability distribution with a fidelity of 93.94%, though the researchers were explicit that this is a proof of principle, a simulation of the logic of life on quantum hardware rather than an account of how life began. [Quantum Biology Society] Where quantum biology has mostly gone looking for quantum mechanical phenomena inside living organisms, this study runs the argument in the opposite direction: it implements the logic of biological evolution inside the physical environment of a quantum computer. The authors describe their line of work as quantum biomimetics, the design of quantum algorithms that imitate biological processes. Researchers at the University of the Basque Country (UPV/EHU) in Bilbao, together with colleagues at several partner institutions, designed the elements of Darwinian natural selection and evolution (self replication, mutation, interaction, and death) as a quantum algorithm that works around the no cloning theorem, one of the founding principles of quantum mechanics. They then succeeded in running it on ibmqx4, the five qubit superconducting chip IBM makes available over the cloud, putting into practice a model the same group had proposed two years earlier. The results, published in Scientific Reports in 2018, stand as the first experimental implementation of a quantum artificial life algorithm on quantum hardware. ■ Two Qubits per Individual: Partial Quantum Cloning Sidesteps the No Cloning Theorem The smallest unit of life in this model, the individual, is built from two qubits, one holding the genotype and one the phenotype. The genotype carries the information that identifies the type of living unit and is handed on from generation to generation. The phenotype is set by the genetic information together with the interaction between the individual and its environment, and that state, along with the information it encodes, is degraded over the individual's lifetime, which is how aging and death enter the model. Self replication, the core of life, is implemented through partial quantum cloning. Quantum mechanics forbids making a perfect copy of an unknown quantum state in full, but it does permit the expectation value of a single chosen observable, here the Pauli Z observable σz, to be written into a blank qubit. Using controlled NOT (CNOT) gates, the researchers passed that particular piece of the genotype's information first into the genotype qubit of the next generation and then into its phenotype qubit, completing a system of heredity in the quantum world. ■ A Practical Compromise with the Hardware: Emulating Dissipation and Mutation Because the flexibility of current quantum hardware is limited, death and mutation had to be simulated by practical approximations. Death should be produced by leaving the phenotype qubit open to its environment so that its information gradually dissipates, but the device could not perform the conditional projection of the phenotype onto the |0⟩ state that this calls for. Instead the researchers applied a sequence of small rotations about the σy axis, one for each simulated time step, to imitate the running out of a lifetime. As they note, this replaces the dissipation with a discrete process and loses its exponential character. Mutation, likewise, was not implemented as a fully random rotation. The researchers fixed the mutation to a single bit flip (σx) and then took a statistical detour: circuits with and without a mutation were run different numbers of times and the data merged afterward so that the intended mutation rate came out of the aggregate. Interaction between individuals is a conditional operation that swaps phenotypes according to the genotypes, and it was built by combining two qubit gates including CNOT, SWAP, and the controlled square root of NOT. ■ The Results, and the Mutation Paradox The experiments proceeded by adding the elements of evolution one after another: interaction between individuals, self replication, dissipation into the environment, and mutation. For the complete model with all four in place, the measured probability distribution matched the ideal prediction with a fidelity of 93.94%. One result along the way was especially curious. When mutation, the element that injects randomness into the system, was included, the fidelity of the data went up rather than down. The researchers give a level headed account of it: mutations tend to homogenize the probability distribution, which is the same effect the errors already present in the experimental platform produce. They also remark on the intriguing parallel with the classical world, where mutation likewise helps a species adapt to changing environments. ■ What Is Actually Quantum Here: Entanglement That Remembers a Lineage The simpler statistical indicators can be reproduced by a classical computer model without much difficulty. Every indicator based on measurements in the σz basis, the researchers point out, can be matched by a classical probability distribution. So which part of this model is genuinely quantum? The answer shows up in the correlations. A partial quantum cloning operation entangles the qubits it involves, carrying the expectation value of σx in the original state into the joint value of σx⊗σx. In a simplified case without mutations or interactions between individuals, measuring σx⊗σx⊗σx⊗σx across the four qubits of two individuals gives α if the two are linked by a self replication event, and α² if they are not causally connected, where α is the expectation value of σx in the precursor genotype. The two histories are therefore distinguishable from the correlation alone. Had the precursor genotype been a classical incoherent mixture instead, that expectation value would be zero and nothing would propagate into these global correlations at all. In other words, without keeping a separate ledger of who descended from whom, as a classical model would have to, the entangled state of the individuals itself records and attests to the lineage. On the chip this signature survived only in part: the measured value of σx⊗σx⊗σx⊗σx was 0.22 against an ideal 0.56, a sizable error. It was nonetheless clearly different from the product of the correlations within each individual taken separately, which comes out at zero, and it is that difference which shows the two individuals to be causally related. ■ Significance and Limits The study is a well executed proof of principle that a microscopic quantum system can hold the grammar of Darwinian evolution consistently, and that it can be run on a real superconducting quantum chip. The paper is explicit, however, that it is not an observation of quantum effects at work in a living organism. The authors leave open the philosophical question of whether the origin of life is genuinely quantum mechanical, and they state that the quantum supremacy referred to in the abstract and conclusion was not achieved in this experiment but remains a target to be explored on larger quantum computers in the future. They also acknowledge that the part of the dynamics carried by σz, which encodes the lifetime and the predator prey character of an individual, is classical information that a simpler classical analogue could predict; what the classical version cannot supply is the entanglement linking the living units. In asking whether the logic of life can be recoded in the language of quantum mechanics, this work sets out another blueprint for where quantum algorithms and the biological sciences may converge. #QuantumBiology #QuantumComputer #IBMQuantumComputer #QuantumArtificialLife #ArtificialLife #Qubit #QuantumEntanglement #NoCloningTheorem #DarwinianEvolution https://www.nature.com/articles/s41598-018-33125-3
By incorporating payoff matrix and population evolution of open systems and proposing density games, pioneering effort has been made to break the following two basic assumptions of evolutionary game theory over the past decades: the payoff from the game determines the reproductive rate of individuals, and the total population size remains constant by a variable called the death rate. On the other hand, most studies on strategic interactions in game theories consider independent individual strategies as the basic unit of analyses. This is in fact not always true in human interaction strategies. It has been unclear that how subjective non-independent species with strategic interactions in open dynamic systems evolve. In this paper, we create the new concept of hybrid quantum games, and propose an open quantum dynamic symbiosis model for exploring the evolution of this type of species under unclosed social settings. We show that the stability conditions of the model with non-independent strategic interactions are influenced by the specific symbiosis states, game payoff matrix and quantum entanglement. The stability conditions of the competition scenario are more stringent than those of the cooperation one. Our system dynamics simulation study demonstrates and verifies the dynamic changes for the population sizes of two different species in the theoretical model, on the basis of the stability conditions of the competition and cooperation scenarios. Our model combines quantum games, game payoff matrix and symbiosis equations organically, and can provide insights into competition, cooperation and symbiosis mechanisms in open systems with subjective non-independent strategies considered.