Quantum Cellular Automata for Quantum Game Of Life

Modelling true Quantum Architecture and Systems has always been next to impossible but with the advent of quantum computing we tend to see some hope, the Qubit structure of a system helps model simulations that run on actual quantum entities, thereby helping model accurate quantum systems.

There have been numerous attempts on mapping classical bits into a true stochastic system, but at best we have achieved a Pseudo-Stochastic System. The problem lies in the initial input, any true stochastic system should evolve over a period of time such that the growth or evolution per se is a function of complexity.

The Conway Game of Life protocol is notoriously one of the most famous ones that simulate the same.

The Protocol goes as follows — The universe of the Game of Life is an infinite, two-dimensional orthogonal grid of square cells, each of which is in one of two possible states, live or dead (or populated and unpopulated, respectively). Every cell interacts with its eight neighbours, which are the cells that are horizontally, vertically, or diagonally adjacent. At each step in time, the following transitions occur:

Any live cell with fewer than two live neighbours dies, as if by underpopulation.

Any live cell with two or three live neighbours lives on to the next generation.

Any live cell with more than three live neighbours dies, as if by overpopulation.

Any dead cell with exactly three live neighbours becomes a live cell, as if by reproduction.

Mathematical Representation

- Let G be the two-dimensional grid representing the Game of Life.

- Let G[i, j] denote the cell at position (i, j) in the grid, where i and j are the row and column indices, respectively.

- Let N(i, j) represent the neighbourhood of the cell G[i, j], consisting of the eight neighbouring cells surrounding it.

Now, we can express the rules of the Game of Life mathematically:

1. Rule 1: Birth

G[i, j] = 1 if G[i, j] = 0 and the number of live neighbours in N(i, j) is exactly 3. Otherwise, G[i, j] remains unchanged.

Mathematically:

G[i, j] = 1 if G[i, j] = 0 and Σ(G[x, y] | (x, y) ∈ N(i, j)) = 3, where Σ denotes the sum of the neighbouring cell values.

2. Rule 2: Survival

G[i, j] = 1 if G[i, j] = 1 and the number of live neighbours in N(i, j) is either 2 or 3. Otherwise, G[i, j] becomes 0.

Mathematically:

G[i, j] = 1 if G[i, j] = 1 and Σ(G[x, y] | (x, y) ∈ N(i, j)) ∈ {2, 3}, otherwise G[i, j] = 0.

3. Rule 3: Death

G[i, j] becomes 0 if G[i, j] = 1 and the number of live neighbours in N(i, j) is less than 2 or greater than 3. Otherwise, G[i, j] remains unchanged.

Mathematically:

G[i, j] = 0 if G[i, j] = 1 and (Σ(G[x, y] | (x, y) ∈ N(i, j)) < 2 or Σ(G[x, y] | (x, y) ∈ N(i, j)) > 3), otherwise G[i, j] remains unchanged.

By applying these rules iteratively to each cell in the grid, we can simulate the evolution of the Game of Life and observe the emergence of various patterns and behaviours.

There are broadly three classes of life, Still Life, Oscillators and Gliders.

1. Still Life:

Still, Life patterns remain unchanged over time and do not exhibit any movement. They are composed of cells that remain in a fixed configuration. Mathematically, we can represent Still Life patterns as follows:

G[i, j] = 1 if G[i, j] = 1 and Σ(G[x, y] | (x, y) ∈ N(i, j)) = 2, where Σ denotes the sum of the neighbouring cell values.

In this case, the sum of the neighbouring cells is always 2, indicating that the cells are surrounded by a stable configuration, and will remain alive.

2. Oscillators:

Oscillators are patterns that exhibit periodic behaviour, repeating their configuration after a certain number of generations. They undergo a cycle of transformations between multiple states. Mathematically, we can represent Oscillators as follows:

G[i, j] = 1 if G[i, j] = 0 and Σ(G[x, y] | (x, y) ∈ N(i, j)) ∈ {2, 3}, where Σ denotes the sum of the neighbouring cell values.

Here, the cells will turn alive if they are currently dead and have either 2 or 3 live neighbours. This rule allows for the oscillation between live and dead states in subsequent generations, creating periodic behaviour.

3. Gliders:

Gliders are patterns that exhibit motion by moving across the grid. They maintain their shape while shifting their position. Mathematically, we represent Gliders by observing their characteristic movement pattern:

G[i, j] = 1 if G[i, j] = 1 and Σ(G[x, y] | (x, y) ∈ N(i, j)) ∈ {2, 3}, where Σ denotes the sum of the neighbouring cell values.

Similar to Oscillators, the rule for Gliders ensures that the cells remain alive if they currently are and have either 2 or 3 live neighbours. This allows for the preservation of the glider’s shape while it moves through subsequent generations.

The intersection of quantum computing and complexity theory has paved the way for groundbreaking research in simulating complex systems. In this article, we explore the synergy between Quantum Cellular Automata (QCA), Quantum Turing Machines (QTM), Universal Turing Machines (UTM), and the development of a Quantum Game of Life protocol. By leveraging the unique properties of quantum mechanics, we aim to create a true stochastic simulation capable of modelling chaos.

The Quantum Game of Life is an extension of Conway’s Game of Life, which is based on classical cellular automata. In the Quantum Game of Life, we introduce quantum principles to enhance computational capabilities and randomness.

To begin, we define a 4x4 matrix as the initial quantum state, where each cell represents a qubit. Similar to Conway’s Game of Life, we apply classical Conway operators, such as birth, survival, and death rules, to the qubits in the quantum matrix. These operators determine the state transitions of the qubits based on the number of neighbouring qubits in specific states.

Additionally, we introduce entanglement correlation operators that generate entanglement between neighbouring qubits. These operators enable qubits to interact and influence each other’s states through entanglement, adding a quantum element to the game.

By iteratively applying the classical Conway operators and entanglement correlation operators, we observe the evolution of the quantum matrix over time. The expanded NxN matrix showcases the true random potential of a Quantum Turing Machine, where the superposition and entanglement of qubits contribute to the emergence of complex patterns and behaviours.

This Quantum Game of Life implementation offers a unique perspective on the interplay between quantum mechanics and computational complexity, providing insights into the potential applications of quantum computing in simulating and understanding complex systems.