Quantum Entanglement Optimization Algorithm (QEO)

Problem Statement

Algorithm Description

Step 1: Graph Representation

• Nodes and Edges: Nodes represent states, and edges represent transitions.
  • A unique identifier for each node.
  • A set of neighboring nodes for each node.
• Multi-dimensional Aspect: The "multi-dimensional" nature can be reflected in various ways:
  • Nodes might possess attributes or coordinates in a higher-dimensional space.
  • Edge costs might depend on multiple factors or dimensions of the interaction.
  • The graph topology itself might represent higher-order relationships (e.g., hyperedges, though the current model simplifies to pairwise edges). The algorithm can accommodate this if node states and cost functions are defined to incorporate these dimensions.
• Dynamic Edge Costs: For a "dynamically changing graph," the cost of transitioning between nodes (edges) may change over time or depend on the overall state of the system. The cost function must be able to query these current costs from the live graph data.
• Quantum State Vector: Each node (or transition) is associated with a quantum state vector. This vector contains complex probability amplitudes related to transitions. The squared magnitude of an amplitude gives the probability of a specific outcome (e.g., taking a particular edge). For simplicity in pseudocode, these are often represented by real numbers (probabilities).

Step 2: Quantum State Initialization

Step 3: Quantum Entanglement

Step 4: Cost Function Calculation

• Classical Cost: Derived from the graph's edge weights. For a dynamically changing graph, the cost function must query the current cost of each transition from the graph's state at the time of evaluation.
• Quantum Component: Aims to quantify the "quality" or "certainty" of a path based on the entanglement and quantum states. It should penalize paths with high entanglement entropy or high uncertainty (e.g., where the probability distribution over next steps is too diffuse), thereby guiding the search towards paths that are not only short classically but also "quantumly preferred" due to lower uncertainty or stronger correlations.

Step 5: Quantum Annealing

Step 6: Path Reconstruction

Step 7: Validation

Pseudocode

def quantum_entanglement_optimization(start_node, target_node, graph):
    # Step 1: Graph Representation
    adjacency_list = create_adjacency_list(graph)

    # Step 2: Quantum State Initialization
    quantum_states = initialize_quantum_states(adjacency_list)

    # Step 3: Quantum Entanglement
    entangled_states = create_entanglement(quantum_states, adjacency_list)

    # Step 4: Cost Function Calculation
    # The cost function needs access to the graph for potentially dynamic costs
    # and the adjacency_list for structure, along with entangled_states for quantum costs.
    # For dynamic costs, 'graph' itself (or a state accessor) would be passed.
    # Here, we assume adjacency_list also contains or can derive current costs if dynamic.
    # If costs are truly dynamic and not captured in initial adjacency_list,
    # define_cost_function would need the main 'graph' object.
    # Let's adjust to pass 'adjacency_list' as the source of costs, assuming it's kept updated
    # or that costs are static for this pseudocode's scope.
    # For a truly dynamic graph, the 'graph' object itself would be more appropriate.
    # We will modify define_cost_function to accept the graph (represented by adjacency_list here for structure and costs).
    cost_function = define_cost_function(entangled_states, adjacency_list, graph) # Pass graph for dynamic costs

    # Step 5: Quantum Annealing
    # The quantum annealing process finds the optimal path.
    optimal_path = quantum_annealing(start_node, target_node, cost_function, entangled_states, adjacency_list)

    # Step 6: Path Reconstruction (Implicit in quantum_annealing's output)
    # The quantum_annealing function is expected to return the optimal path directly.

    # Step 7: Validation
    # Pass the original 'graph' object for current cost validation.
    validate_path(optimal_path, graph, adjacency_list)
    # In a more complex version, the result of validate_path could trigger
    # adjustments and a re-run of the annealing process if the path is not satisfactory.
    # This loop is omitted here for pseudocode simplicity.

    return optimal_path

# Import necessary modules for pseudocode clarity
import random
import math
import collections # For BFS in initial path generation

def create_adjacency_list(graph):
    # This function creates an adjacency list representing the graph's structure
    # and costs *at the time of creation*. If the graph is dynamic,
    # the costs stored here might become stale. The cost_function should
    # ideally refer to the live 'graph' object for current costs.
    adjacency_list = {}
    for node in graph:
        adjacency_list[node] = []
        # Ensure graph[node] is treated as a dictionary for iteration
        if isinstance(graph[node], dict):
            for neighbor, cost in graph[node].items():
                adjacency_list[node].append((neighbor, cost))
        else: # Handle cases where graph[node] might be a list of tuples or similar
            for item in graph[node]:
                if isinstance(item, tuple) and len(item) == 2:
                    adjacency_list[node].append(item)
                else:
                    # Attempt to parse if it's just a neighbor name, assume default cost
                    adjacency_list[node].append((item, 1)) # Default cost if not specified
    return adjacency_list

def initialize_quantum_states(adjacency_list):
    quantum_states = {}
    for node, neighbors in adjacency_list.items():
        # Initialize quantum state vector for each node.
        # This vector represents complex probability amplitudes.
        # For simplicity in pseudocode, we use real numbers representing uniform probabilities
        # as a proxy for normalized squared magnitudes of amplitudes.
        if neighbors:
            # Each element could be 1/sqrt(len(neighbors)) if these were actual amplitudes for a uniform superposition.
            # Or, if these are probabilities directly:
            quantum_states[node] = [1.0 / len(neighbors)] * len(neighbors)
        else:
            quantum_states[node] = [] # No transitions if no neighbors
    return quantum_states

def create_entanglement(quantum_states, adjacency_list):
    entangled_states = {}
    for node, neighbors in adjacency_list.items():
        entangled_states[node] = []
        for neighbor_info in neighbors:
            neighbor_node = neighbor_info[0] # Extract neighbor node from (neighbor, cost) tuple
            # Create entanglement between neighboring nodes.
            # For pseudocode, this means associating the neighbor's quantum state.
            # In a real quantum system, this would involve quantum gates.
            if neighbor_node in quantum_states:
                entangled_states[node].append((neighbor_node, quantum_states[neighbor_node]))
            else:
                # Handle cases where a neighbor might not have its own quantum state initialized yet
                entangled_states[node].append((neighbor_node, [])) # Placeholder for uninitialized state
    return entangled_states

# Pass the main 'graph_data' (or a graph accessor object) for dynamic cost retrieval.
# The 'adjacency_list' can still be used for graph structure if it's separate from dynamic cost data.
# For this pseudocode, we'll assume 'graph_data' is the primary source for costs if they are dynamic.
# If costs are static, 'adjacency_list' might be sufficient.
# 'graph_source_for_costs' is the primary graph object, potentially dynamic.
# 'adjacency_list_structure' is the initial structural representation, can be used for topology checks
# if 'graph_source_for_costs' doesn't explicitly list all non-edges.
def define_cost_function(entangled_states, adjacency_list_structure, graph_source_for_costs):
    def cost_function(path):
        total_cost = 0
        for i in range(len(path) - 1):
            current_node = path[i]
            next_node = path[i + 1]

            classical_cost_value = 0
            classical_cost_found = False
            
            # Retrieve classical cost for the transition current_node -> next_node.
            # Prioritize the 'graph_source_for_costs' for dynamic cost updates.
            if isinstance(graph_source_for_costs, dict) and \
               current_node in graph_source_for_costs and \
               isinstance(graph_source_for_costs[current_node], dict) and \
               next_node in graph_source_for_costs[current_node]:
                
                classical_cost_value = graph_source_for_costs[current_node][next_node]
                total_cost += classical_cost_value
                classical_cost_found = True
            else:
                # Fallback or error: If the path implies an edge that doesn't exist
                # in the current dynamic graph_source_for_costs, it's an invalid transition.
                # We can also check adjacency_list_structure for the *existence* of the edge
                # if graph_source_for_costs might only store costs for existing edges.
                # For simplicity, if not in graph_source_for_costs, assume invalid or use adj_list for structure.
                edge_exists_in_structure = False
                for neighbor_node_in_struct, _ in adjacency_list_structure.get(current_node, []):
                    if neighbor_node_in_struct == next_node:
                        edge_exists_in_structure = True
                        break
                if not edge_exists_in_structure: # Edge doesn't even exist structurally
                    total_cost += float('inf') 
                    continue # Invalid path segment
                else:
                    # Edge exists structurally, but no cost in dynamic graph_source (or it's not found)
                    # This case implies an issue or that the cost is implicitly zero or infinite.
                    # For this pseudocode, we'll treat it as an error / infinite cost if not explicitly found.
                    total_cost += float('inf') # Or handle as per specific problem definition
                    continue

            # Calculate quantum cost component for the transition current_node -> next_node
            # This component should reflect the algorithm's preference for paths with
            # "low quantum uncertainty" or specific entanglement characteristics.
            # The current pseudocode uses a placeholder. A true quantum cost would
            # derive from the properties of the `entangled_states` for the transition.
            quantum_cost_value = 0
            # Search for the specific transition in the entangled_states
            # to derive its quantum cost contribution.
            if current_node in entangled_states:
                for entangled_neighbor_node, neighbor_quantum_state_vector in entangled_states[current_node]:
                    if entangled_neighbor_node == next_node:
                        # Placeholder for quantum cost calculation:
                        # In a real QEO, this would involve a metric derived from
                        # `neighbor_quantum_state_vector` (e.g., related to its entropy,
                        # stability, or specific amplitude patterns).
                        # For this pseudocode, we use a simple proxy:
                        # a larger number of components in the state vector might imply higher
                        # "uncertainty" or "complexity" for that transition.
                        if neighbor_quantum_state_vector: # Check if the state vector is not empty
                            quantum_cost_value = len(neighbor_quantum_state_vector) * 0.1 # Example proxy
                        break # Found the relevant entangled neighbor
            total_cost += quantum_cost_value
        return total_cost
    return cost_function

def quantum_annealing(start_node, target_node, cost_function, entangled_states, adjacency_list):
    # --- IMPORTANT NOTE ON THIS PSEUDOCODE ---
    # This function implements *classical simulated annealing* as a conceptual stand-in
    # for true quantum annealing. Representing the dynamics of a quantum annealer
    # (e.g., evolving a Hamiltonian, managing quantum fluctuations) is beyond the scope
    # of simple procedural pseudocode. A real quantum annealing process would operate
    # on quantum bits (qubits) and leverage quantum phenomena like superposition and tunneling.
    # The `entangled_states` parameter is passed but not deeply utilized in a quantum
    # mechanical way by this classical simulation, serving more as a placeholder for where
    # quantum information would be critical.
    #
    # Note on Dynamic Structure: This pseudocode's BFS initialization and path perturbation
    # rely on the `adjacency_list` passed, which reflects the graph structure at the time
    # of its creation. If the graph's *structure* (nodes/edges) is highly dynamic *during*
    # the annealing process itself, this `adjacency_list` would become stale. A more robust
    # implementation for such scenarios would need to query the live graph state for
    # structural information repeatedly or use a graph representation that updates.
    # The cost_function, however, *is* designed to use live costs from the graph object.
    #
    # In a true quantum annealing setup:
    # 1. The problem (finding an optimal path) would be mapped to an Ising model Hamiltonian.
    # 2. An initial Hamiltonian (often with strong transverse fields) allows exploration of many states (superposition).
    # 3. The system evolves adiabatically, gradually changing the Hamiltonian to one whose ground state
    #    encodes the solution to the optimization problem.
    # 4. The final state is measured to obtain the optimal path.

    # Classical Simulated Annealing Parameters:
    num_iterations = 1000
    temperature = 1.0
    cooling_rate = 0.99

    # Step 1: Initialize with a valid path (e.g., using Breadth-First Search)
    # This ensures the initial path is connected from start to target.
    queue = collections.deque([(start_node, [start_node])])
    visited = {start_node}
    current_path = []

    while queue:
        node, path = queue.popleft()
        if node == target_node:
            current_path = path
            break
        for neighbor_info, _ in adjacency_list.get(node, []):
            neighbor_node = neighbor_info
            if neighbor_node not in visited:
                visited.add(neighbor_node)
                queue.append((neighbor_node, path + [neighbor_node]))
    
    if not current_path:
        # If no path found by BFS, target is unreachable or graph is disconnected.
        # Return an empty path or handle as an error.
        return [] # Or raise an exception

    current_cost = cost_function(current_path)

    # Step 2: Iteratively perturb the path and apply annealing logic
    for _ in range(num_iterations):
        # Generate a new candidate path by making a small, valid perturbation.
        # This perturbation should maintain connectivity from start to target.
        # This is a conceptual representation of how quantum fluctuations or
        # classical annealing perturbations would explore the solution space.

        new_path = list(current_path) # Start with a copy of the current path

        # --- Path Perturbation Strategy (Conceptual for Pseudocode) ---
        # In a real simulated annealing or quantum annealing implementation,
        # the perturbation strategy is crucial for effective exploration of the
        # solution space while maintaining path validity. Common strategies for
        # pathfinding involve:
        # 1. Selecting two random distinct indices i, j in the current_path (i < j).
        # 2. Finding a new valid sub-path between current_path[i] and current_path[j]
        #    (e.g., using a limited Breadth-First Search, Depth-First Search, or A*).
        # 3. Replacing the segment current_path[i...j] with this new sub-path.
        # This ensures the path remains connected from start to target.
        # For simplicity in this pseudocode, we abstract this complex process
        # and use a highly simplified perturbation example.
        
        # Example of a highly simplified perturbation (conceptual):
        # This example attempts to reroute a small segment of the path.
        # In a real implementation, robust validity checks and more sophisticated
        # sub-path generation would be required.
        if len(current_path) > 2:
            # Pick a random intermediate node (not start or end)
            idx_to_perturb = random.randint(1, len(current_path) - 2)
            node_before = current_path[idx_to_perturb - 1]
            node_after = current_path[idx_to_perturb + 1]
            
            # Find alternative paths between node_before and node_after
            # For pseudocode, we'll just try to insert a random neighbor if it connects.
            possible_intermediate_nodes = [
                n for n, _ in adjacency_list.get(node_before, [])
                if n != node_after and n not in current_path # Avoid immediate back-and-forth or existing path
            ]
            
            if possible_intermediate_nodes:
                # Try to find a path from the new intermediate node to node_after
                # This is a simplified local search.
                new_intermediate = random.choice(possible_intermediate_nodes)
                
                # Check if new_intermediate can reach node_after
                # (e.g., direct edge or short path)
                can_reach_after = False
                for n_info, _ in adjacency_list.get(new_intermediate, []):
                    if n_info == node_after:
                        can_reach_after = True
                        break
                
                if can_reach_after:
                    # Construct the new path with the rerouted segment
                    new_path = current_path[:idx_to_perturb] + [new_intermediate, node_after] + current_path[idx_to_perturb + 2:]
                    # Note: In a real implementation, ensuring the new path's
                    # validity and connectivity after perturbation is critical.
                else:
                    # If rerouting failed, keep the old path for this iteration
                    new_path = current_path
            else:
                new_path = current_path # No valid perturbation found for this iteration
        else:
            new_path = current_path # Path too short for this type of perturbation

        # Calculate the cost of the new path
        new_cost = cost_function(new_path)

        # Acceptance probability (Metropolis-Hastings criterion)
        # Accept if new path is better, or with a probability if it's worse
        if new_cost < current_cost or (temperature > 0 and random.random() < math.exp((current_cost - new_cost) / temperature)):
            current_path = new_path
            current_cost = new_cost

        # Reduce the temperature (cooling schedule)
        temperature *= cooling_rate

    # --- NOTE ON PATH RECONSTRUCTION ---
    # In this classical simulation, `current_path` directly holds the best path found.
    # In a true quantum annealing scenario, after the annealing schedule, one would
    # perform measurements on the quantum system (qubits) to determine the final state,
    # from which the optimal path would be reconstructed. This measurement process
    # collapses the quantum superposition into a classical result.

    return current_path

# The reconstruct_path function is no longer needed as quantum_annealing returns the path directly.

# 'graph_source_for_costs' is the live graph data for current costs.
# 'adjacency_list_structure' is the initial graph structure.
def validate_path(path, graph_source_for_costs, adjacency_list_structure):
    total_cost = 0
    path_valid = True

    if not path:
        print("Path is empty or invalid.")
        return 0, False # Cost, isValid

    for i in range(len(path) - 1):
        current_node = path[i]
        next_node = path[i + 1]

        current_transition_cost = 0
        transition_found_in_dynamic_graph = False

        # Check for the transition and its cost in the current state of the graph
        if isinstance(graph_source_for_costs, dict) and \
           current_node in graph_source_for_costs and \
           isinstance(graph_source_for_costs[current_node], dict) and \
           next_node in graph_source_for_costs[current_node]:
            
            current_transition_cost = graph_source_for_costs[current_node][next_node]
            total_cost += current_transition_cost
            transition_found_in_dynamic_graph = True
        
        if not transition_found_in_dynamic_graph:
            # If not found in dynamic graph, check if the edge even exists structurally
            # based on the initial adjacency_list_structure.
            edge_exists_structurally = False
            for neighbor_node_in_struct, _ in adjacency_list_structure.get(current_node, []):
                if neighbor_node_in_struct == next_node:
                    edge_exists_structurally = True
                    break
            
            if edge_exists_structurally:
                # Edge exists in structure, but not in current dynamic graph (or cost is missing)
                # This implies the edge might have been removed or its cost became undefined.
                print(f"Transition {current_node} -> {next_node} exists structurally but not in current dynamic graph state or has no cost.")
                total_cost = float('inf') # Or handle as per specific problem definition
                path_valid = False
                break
            else:
                # Edge doesn't exist even in the basic structure.
                print(f"Invalid transition (edge does not exist): {current_node} -> {next_node}")
                total_cost = float('inf')
                path_valid = False
                break

    if path_valid:
        print(f"Total cost of the path (using current graph costs): {total_cost}")
        print("Path is valid according to current graph state.")
    else:
        print(f"Path is invalid or has infinite cost based on current graph state. Calculated cost: {total_cost}")

    return total_cost, path_valid

Complexity Analysis

• The number of nodes and edges in the graph
• The dimensionality of the graph
• The complexity of the quantum state initialization and entanglement processes
• The complexity of the cost function calculation
• The number of iterations required for the quantum annealing process

Applications

• Optimization of complex networks: The algorithm can be used to optimize routing in communication networks, logistics networks, and transportation networks.
• Quantum computing and quantum information processing: The QEO algorithm can be applied to optimize quantum circuits and quantum algorithms, improving their efficiency and performance.
• Machine learning and artificial intelligence: The algorithm can be used to optimize neural networks, decision trees, and other machine learning models, leading to improved accuracy and efficiency.
• Cryptography and secure communication: The QEO algorithm can be used to optimize encryption algorithms and secure communication protocols, enhancing their security and efficiency.
• Financial modeling and risk management: The algorithm can be used to optimize portfolio management, risk assessment, and financial forecasting, leading to improved decision-making and risk management.