Parrondos 1d game

Author: jishnurajendran

parrondos_qwalk_1D.py

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+# %%
+import numpy as np
+import matplotlib.pyplot as plt
+from quantumwalks import run_quantum_walk_simulation
+from quantumwalks.discrete_walk import QuantumWalk
+from quantumwalks.utils.operators import create_shift_operators
+
+def create_custom_coin(alpha: float, beta: float, gamma: float) -> np.ndarray:
+    """
+    Create a custom coin operator U(α,β,γ) using vectorized operations.
+    """
+    return np.array([
+        [np.exp(1j * alpha) * np.cos(beta), -np.exp(-1j * gamma) * np.sin(beta)],
+        [np.exp(1j * gamma) * np.sin(beta), np.exp(-1j * alpha) * np.cos(beta)]
+    ], dtype=np.complex128)
+
+def calculate_lr_difference_alternating_coins(qw: QuantumWalk, coin_A: np.ndarray, coin_B: np.ndarray) -> np.ndarray:
+    """
+    Calculate P_L - P_R for each step of the walk using alternating coins A and B.
+    """
+    # Initialize state
+    posn0 = np.zeros(qw.P)
+    posn0[qw.N] = 1
+    psi = np.kron(posn0, qw.initial_coin_state)
+
+    # Pre-allocate arrays
+    differences = np.zeros(qw.N + 1)
+    positions = np.arange(-qw.N, qw.N + 1)
+
+    # Create measurement operators matrix once
+    measurement_ops = np.array([
+        np.kron(np.outer(np.eye(qw.P)[i], np.eye(qw.P)[i]), np.eye(2))
+        for i in range(qw.P)
+    ])
+
+    # Create shift operators using the utility function
+    shift_plus, shift_minus = create_shift_operators(qw.P)
+    shift_op = (np.kron(shift_plus, np.outer(qw.coin0, qw.coin0)) +
+                np.kron(shift_minus, np.outer(qw.coin1, qw.coin1)))
+
+    # Calculate evolution operators for both coins
+    U_A = shift_op @ np.kron(np.eye(qw.P), coin_A)
+    U_B = shift_op @ np.kron(np.eye(qw.P), coin_B)
+
+    # Calculate for each step
+    for step in range(qw.N + 1):
+        # Vectorized probability calculation
+        projections = measurement_ops @ psi
+        prob = np.real(np.sum(projections * np.conjugate(projections), axis=1))
+
+        # Vectorized P_L and P_R calculation
+        P_L = np.sum(prob[positions < 0])
+        P_R = np.sum(prob[positions >= 0])
+        differences[step] = P_L - P_R
+
+        # Evolve state with alternating coin operators
+        if step < qw.N:
+            if step % 2 == 0:
+                psi = U_A @ psi  # Apply coin A
+            else:
+                psi = U_B @ psi  # Apply coin B
+
+    return differences, prob  # Return both differences and final probability distribution
+
+def run_simulation_alternating_coins(steps: int, 
+                                   alpha_A: float, beta_A: float, gamma_A: float,
+                                   alpha_B: float, beta_B: float, gamma_B: float):
+    """
+    Run simulation with alternating coins A and B.
+    """
+    # Create both coin operators
+    coin_A = create_custom_coin(alpha_A, beta_A, gamma_A)
+    coin_B = create_custom_coin(alpha_B, beta_B, gamma_B)
+    initial_state = np.array([1, -1], dtype=np.complex128) / np.sqrt(2)
+
+    print(f"Running quantum walk simulation with alternating coins:")
+    print(f"Coin A parameters: α={alpha_A:.2f}, β={beta_A:.2f}, γ={gamma_A:.2f}")
+    print(f"Coin B parameters: α={alpha_B:.2f}, β={beta_B:.2f}, γ={gamma_B:.2f}")
+
+    # Create quantum walk object (using coin_A as default)
+    qw = QuantumWalk(steps, initial_state, coin_A)
+
+    # Calculate differences and probabilities using alternating coins
+    differences, prob = calculate_lr_difference_alternating_coins(qw, coin_A, coin_B)
+
+    # Plot results
+    plot_results(qw, prob, 0.0, differences, steps)  # Using 0.0 as execution time placeholder
+
+    return qw, differences, prob
+
+def plot_results(qw: QuantumWalk, prob: np.ndarray,
+                exec_time: float, differences: np.ndarray, steps: int):
+    """
+    Plot both probability distribution and P_L - P_R differences.
+    """
+    # Plot probability distribution
+    positions = np.arange(-qw.N, qw.N + 1)
+    
+    plt.figure(figsize=(12, 8))
+    plt.plot(positions, prob, 'b-', label='Probability', linewidth=1.5)
+    plt.plot(positions, prob, 'ro', markersize=4)
+    plt.fill_between(positions, prob, alpha=0.3)
+    plt.grid(True, alpha=0.3)
+    plt.xlabel('Position', fontsize=12)
+    plt.ylabel('Probability', fontsize=12)
+    plt.title(f'Quantum Walk Distribution after {steps} steps\n(Alternating Coins)', fontsize=14)
+    plt.legend()
+    plt.tight_layout()
+    plt.show()
+
+    # Plot P_L - P_R differences
+    plt.figure(figsize=(12, 8))
+    x = np.arange(steps + 1)
+    plt.plot(x, differences, 'b-', label='P_L - P_R', linewidth=1.5)
+    plt.plot(x, differences, 'ro', markersize=4)
+
+    # Set y-axis range from -1 to 1
+    plt.ylim(-1, 1)
+
+    plt.grid(True, alpha=0.3)
+    plt.xlabel('Step', fontsize=12)
+    plt.ylabel('P_L - P_R', fontsize=12)
+    plt.title('Difference between Left and Right Probabilities\n(Alternating Coins)', fontsize=14)
+    plt.legend()
+    plt.tight_layout()
+    plt.show()
+
+
+# %%
+
+if __name__ == "__main__":
+    # Set parameters
+    steps = 800
+    
+    # Parameters for Coin A
+    alpha_A = -51.0/180 * np.pi  # Converting to radians
+    beta_A = 45.0/180 * np.pi
+    gamma_A = 0.0/180 * np.pi
+    
+    # Parameters for Coin B
+    alpha_B = 0.0/180 * np.pi
+    beta_B = 88.0/180 * np.pi
+    gamma_B = -16.0/180 * np.pi
+
+    # Run simulation with alternating coins
+    qw, differences, prob = run_simulation_alternating_coins(
+        steps, 
+        alpha_A, beta_A, gamma_A,
+        alpha_B, beta_B, gamma_B
+    )
+
+    # Print some statistics
+    positions = np.arange(-qw.N, qw.N + 1)
+    mean_pos = np.average(positions, weights=prob)
+    std_dev = np.sqrt(np.average((positions - mean_pos)**2, weights=prob))
+    
+    print("\nFinal Statistics:")
+    print(f"Mean Position: {mean_pos:.4f}")
+    print(f"Standard Deviation: {std_dev:.4f}")
+    print(f"Max Probability: {np.max(prob):.4f}")
+
+# %%
+
+
+