object

defmodule Object.QuantumAlgorithms do
  @moduledoc """
  Quantum-inspired algorithms for AAOS optimization and computation.
  
  Implements quantum computational paradigms on classical hardware including:
  
  - Quantum-inspired evolutionary algorithms (QIEA)
  - Variational quantum eigensolvers (VQE) simulation
  - Quantum approximate optimization algorithm (QAOA)
  - Quantum neural networks (QNN) with parameterized quantum circuits
  - Quantum reinforcement learning algorithms
  - Quantum annealing simulation for combinatorial optimization
  - Quantum walks for graph problems
  - Tensor network methods for many-body systems
  - Quantum error correction codes for robust computation
  """
  
  use GenServer
  require Logger
  
  # Quantum System Constants
  @qubit_dimensions 2
  @max_qubits 64
  @circuit_depth 100
  @measurement_shots 8192
  @entanglement_threshold 0.8
  
  # Optimization Constants
  @qaoa_layers 10
  @vqe_iterations 1000
  @annealing_steps 10000
  @population_size 100
  @mutation_rate 0.01
  
  @type qubit :: %{
    amplitude_0: Complex.t(),
    amplitude_1: Complex.t(),
    phase: float(),
    entangled_with: [non_neg_integer()]
  }
  
  @type quantum_state :: %{
    qubits: [qubit()],
    global_phase: float(),
    entanglement_matrix: [[float()]],
    measurement_basis: :computational | :hadamard | :bell
  }
  
  @type quantum_gate :: %{
    type: :pauli_x | :pauli_y | :pauli_z | :hadamard | :cnot | :rotation | :custom,
    target_qubits: [non_neg_integer()],
    parameters: [float()],
    unitary_matrix: [[Complex.t()]]
  }
  
  @type quantum_circuit :: %{
    gates: [quantum_gate()],
    qubit_count: non_neg_integer(),
    depth: non_neg_integer(),
    parameters: [float()],
    cost_function: function()
  }
  
  @type optimization_problem :: %{
    objective_function: function(),
    constraints: [function()],
    variable_bounds: [{float(), float()}],
    problem_type: :minimization | :maximization,
    classical_solution: term() | nil
  }
  
  @type quantum_individual :: %{
    chromosome: [float()],
    quantum_state: quantum_state(),
    fitness: float(),
    entanglement_score: float(),
    generation: non_neg_integer()
  }
  
  @type state :: %{
    quantum_systems: %{binary() => quantum_state()},
    optimization_problems: %{binary() => optimization_problem()},
    active_circuits: %{binary() => quantum_circuit()},
    populations: %{binary() => [quantum_individual()]},
    performance_metrics: %{
      convergence_rate: float(),
      quantum_advantage: float(),
      entanglement_utilization: float()
    },
    hardware_simulation: %{
      noise_model: map(),
      error_rates: map(),
      decoherence_times: map()
    }
  }
  
  # Client API
  
  @doc """
  Starts the quantum algorithms service.
  """
  def start_link(opts \\ []) do
    GenServer.start_link(__MODULE__, opts, name: __MODULE__)
  end
  
  @doc """
  Initializes a quantum system with specified number of qubits.
  """
  @spec initialize_quantum_system(binary(), non_neg_integer()) :: :ok | {:error, term()}
  def initialize_quantum_system(system_id, num_qubits) do
    GenServer.call(__MODULE__, {:init_quantum_system, system_id, num_qubits})
  end
  
  @doc """
  Applies a quantum gate to the specified system.
  """
  @spec apply_quantum_gate(binary(), quantum_gate()) :: :ok | {:error, term()}
  def apply_quantum_gate(system_id, gate) do
    GenServer.call(__MODULE__, {:apply_gate, system_id, gate})
  end
  
  @doc """
  Runs QAOA (Quantum Approximate Optimization Algorithm) for combinatorial problems.
  """
  @spec run_qaoa(optimization_problem(), non_neg_integer()) :: {:ok, term()} | {:error, term()}
  def run_qaoa(problem, num_layers \\ @qaoa_layers) do
    GenServer.call(__MODULE__, {:run_qaoa, problem, num_layers}, 60000)
  end
  
  @doc """
  Executes VQE (Variational Quantum Eigensolver) for finding ground states.
  """
  @spec run_vqe(function(), quantum_circuit()) :: {:ok, {float(), [float()]}} | {:error, term()}
  def run_vqe(hamiltonian, ansatz_circuit) do
    GenServer.call(__MODULE__, {:run_vqe, hamiltonian, ansatz_circuit}, 60000)
  end
  
  @doc """
  Performs quantum-inspired evolutionary algorithm optimization.
  """
  @spec quantum_evolutionary_algorithm(optimization_problem(), map()) :: {:ok, term()} | {:error, term()}
  def quantum_evolutionary_algorithm(problem, options \\ %{}) do
    GenServer.call(__MODULE__, {:qiea, problem, options}, 120000)
  end
  
  @doc """
  Simulates quantum annealing for combinatorial optimization.
  """
  @spec quantum_annealing(optimization_problem(), non_neg_integer()) :: {:ok, term()} | {:error, term()}
  def quantum_annealing(problem, steps \\ @annealing_steps) do
    GenServer.call(__MODULE__, {:quantum_annealing, problem, steps}, 120000)
  end
  
  @doc """
  Performs quantum walk on a graph structure.
  """
  @spec quantum_walk(map(), non_neg_integer(), non_neg_integer()) :: {:ok, [float()]} | {:error, term()}
  def quantum_walk(graph, starting_node, num_steps) do
    GenServer.call(__MODULE__, {:quantum_walk, graph, starting_node, num_steps})
  end
  
  @doc """
  Trains a quantum neural network with parameterized quantum circuits.
  """
  @spec train_quantum_neural_network([{term(), term()}], quantum_circuit(), map()) :: 
    {:ok, quantum_circuit()} | {:error, term()}
  def train_quantum_neural_network(training_data, initial_circuit, options \\ %{}) do
    GenServer.call(__MODULE__, {:train_qnn, training_data, initial_circuit, options}, 180000)
  end
  
  @doc """
  Generates quantum error correction codes.
  """
  @spec generate_qec_code(non_neg_integer(), non_neg_integer(), non_neg_integer()) :: 
    {:ok, map()} | {:error, term()}
  def generate_qec_code(data_qubits, syndrome_qubits, distance) do
    GenServer.call(__MODULE__, {:generate_qec, data_qubits, syndrome_qubits, distance})
  end
  
  # Server Callbacks
  
  @impl true
  def init(opts) do
    # Initialize quantum simulation environment
    state = %{
      quantum_systems: %{},
      optimization_problems: %{},
      active_circuits: %{},
      populations: %{},
      performance_metrics: %{
        convergence_rate: 0.0,
        quantum_advantage: 0.0,
        entanglement_utilization: 0.0
      },
      hardware_simulation: %{
        noise_model: initialize_noise_model(),
        error_rates: initialize_error_rates(),
        decoherence_times: initialize_decoherence_times()
      }
    }
    
    Logger.info("Quantum algorithms service initialized with #{@max_qubits}-qubit simulation capacity")
    {:ok, state}
  end
  
  @impl true
  def handle_call({:init_quantum_system, system_id, num_qubits}, _from, state) do
    if num_qubits > @max_qubits do
      {:reply, {:error, :too_many_qubits}, state}
    else
      quantum_system = initialize_quantum_state(num_qubits)
      new_systems = Map.put(state.quantum_systems, system_id, quantum_system)
      new_state = %{state | quantum_systems: new_systems}
      {:reply, :ok, new_state}
    end
  end
  
  @impl true
  def handle_call({:apply_gate, system_id, gate}, _from, state) do
    case Map.get(state.quantum_systems, system_id) do
      nil ->
        {:reply, {:error, :system_not_found}, state}
      quantum_system ->
        case apply_gate_to_system(gate, quantum_system) do
          {:ok, new_system} ->
            new_systems = Map.put(state.quantum_systems, system_id, new_system)
            new_state = %{state | quantum_systems: new_systems}
            {:reply, :ok, new_state}
          error ->
            {:reply, error, state}
        end
    end
  end
  
  @impl true
  def handle_call({:run_qaoa, problem, num_layers}, _from, state) do
    case execute_qaoa_algorithm(problem, num_layers, state) do
      {:ok, result} ->
        {:reply, {:ok, result}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:run_vqe, hamiltonian, ansatz_circuit}, _from, state) do
    case execute_vqe_algorithm(hamiltonian, ansatz_circuit, state) do
      {:ok, {energy, parameters}} ->
        {:reply, {:ok, {energy, parameters}}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:qiea, problem, options}, _from, state) do
    case execute_quantum_evolutionary_algorithm(problem, options, state) do
      {:ok, result, new_populations} ->
        new_state = %{state | populations: Map.merge(state.populations, new_populations)}
        {:reply, {:ok, result}, new_state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:quantum_annealing, problem, steps}, _from, state) do
    case execute_quantum_annealing(problem, steps, state) do
      {:ok, result} ->
        {:reply, {:ok, result}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:quantum_walk, graph, starting_node, num_steps}, _from, state) do
    case execute_quantum_walk(graph, starting_node, num_steps, state) do
      {:ok, probability_distribution} ->
        {:reply, {:ok, probability_distribution}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:train_qnn, training_data, initial_circuit, options}, _from, state) do
    case train_quantum_neural_network_impl(training_data, initial_circuit, options, state) do
      {:ok, trained_circuit} ->
        {:reply, {:ok, trained_circuit}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  @impl true
  def handle_call({:generate_qec, data_qubits, syndrome_qubits, distance}, _from, state) do
    case generate_quantum_error_correction_code(data_qubits, syndrome_qubits, distance) do
      {:ok, qec_code} ->
        {:reply, {:ok, qec_code}, state}
      error ->
        {:reply, error, state}
    end
  end
  
  # Quantum State Initialization and Manipulation
  
  defp initialize_quantum_state(num_qubits) do
    qubits = for _ <- 1..num_qubits do
      %{
        amplitude_0: Complex.new(1.0, 0.0),  # |0⟩ state
        amplitude_1: Complex.new(0.0, 0.0),  # |1⟩ state
        phase: 0.0,
        entangled_with: []
      }
    end
    
    entanglement_matrix = for _ <- 1..num_qubits do
      for _ <- 1..num_qubits, do: 0.0
    end
    
    %{
      qubits: qubits,
      global_phase: 0.0,
      entanglement_matrix: entanglement_matrix,
      measurement_basis: :computational
    }
  end
  
  defp apply_gate_to_system(gate, quantum_system) do
    try do
      case gate.type do
        :hadamard ->
          apply_hadamard_gate(gate.target_qubits, quantum_system)
        :pauli_x ->
          apply_pauli_x_gate(gate.target_qubits, quantum_system)
        :pauli_y ->
          apply_pauli_y_gate(gate.target_qubits, quantum_system)
        :pauli_z ->
          apply_pauli_z_gate(gate.target_qubits, quantum_system)
        :cnot ->
          apply_cnot_gate(gate.target_qubits, quantum_system)
        :rotation ->
          apply_rotation_gate(gate.target_qubits, gate.parameters, quantum_system)
        :custom ->
          apply_custom_gate(gate.unitary_matrix, gate.target_qubits, quantum_system)
        _ ->
          {:error, :unsupported_gate_type}
      end
    catch
      error -> {:error, error}
    end
  end
  
  defp apply_hadamard_gate([target_qubit], quantum_system) do
    qubits = quantum_system.qubits
    target = Enum.at(qubits, target_qubit)
    
    # H|0⟩ = (|0⟩ + |1⟩)/√2, H|1⟩ = (|0⟩ - |1⟩)/√2
    sqrt_2_inv = 1.0 / :math.sqrt(2.0)
    
    new_amp_0 = Complex.multiply(
      Complex.add(target.amplitude_0, target.amplitude_1),
      Complex.new(sqrt_2_inv, 0.0)
    )
    new_amp_1 = Complex.multiply(
      Complex.subtract(target.amplitude_0, target.amplitude_1),
      Complex.new(sqrt_2_inv, 0.0)
    )
    
    new_target = %{target | amplitude_0: new_amp_0, amplitude_1: new_amp_1}
    new_qubits = List.replace_at(qubits, target_qubit, new_target)
    
    {:ok, %{quantum_system | qubits: new_qubits}}
  end
  
  defp apply_pauli_x_gate([target_qubit], quantum_system) do
    qubits = quantum_system.qubits
    target = Enum.at(qubits, target_qubit)
    
    # X|0⟩ = |1⟩, X|1⟩ = |0⟩ (bit flip)
    new_target = %{target | 
      amplitude_0: target.amplitude_1,
      amplitude_1: target.amplitude_0
    }
    
    new_qubits = List.replace_at(qubits, target_qubit, new_target)
    {:ok, %{quantum_system | qubits: new_qubits}}
  end
  
  defp apply_pauli_y_gate([target_qubit], quantum_system) do
    qubits = quantum_system.qubits
    target = Enum.at(qubits, target_qubit)
    
    # Y|0⟩ = i|1⟩, Y|1⟩ = -i|0⟩
    i = Complex.new(0.0, 1.0)
    neg_i = Complex.new(0.0, -1.0)
    
    new_target = %{target |
      amplitude_0: Complex.multiply(neg_i, target.amplitude_1),
      amplitude_1: Complex.multiply(i, target.amplitude_0)
    }
    
    new_qubits = List.replace_at(qubits, target_qubit, new_target)
    {:ok, %{quantum_system | qubits: new_qubits}}
  end
  
  defp apply_pauli_z_gate([target_qubit], quantum_system) do
    qubits = quantum_system.qubits
    target = Enum.at(qubits, target_qubit)
    
    # Z|0⟩ = |0⟩, Z|1⟩ = -|1⟩ (phase flip)
    new_target = %{target |
      amplitude_1: Complex.multiply(target.amplitude_1, Complex.new(-1.0, 0.0))
    }
    
    new_qubits = List.replace_at(qubits, target_qubit, new_target)
    {:ok, %{quantum_system | qubits: new_qubits}}
  end
  
  defp apply_cnot_gate([control_qubit, target_qubit], quantum_system) do
    qubits = quantum_system.qubits
    control = Enum.at(qubits, control_qubit)
    target = Enum.at(qubits, target_qubit)
    
    # CNOT: if control is |1⟩, flip target
    control_prob_1 = Complex.multiply(control.amplitude_1, Complex.conjugate(control.amplitude_1)) |> Complex.real()
    
    if control_prob_1 > 0.5 do
      # Apply X gate to target
      new_target = %{target |
        amplitude_0: target.amplitude_1,
        amplitude_1: target.amplitude_0
      }
      
      # Update entanglement
      new_entanglement = update_entanglement_matrix(
        quantum_system.entanglement_matrix,
        control_qubit,
        target_qubit
      )
      
      new_qubits = qubits
      |> List.replace_at(target_qubit, new_target)
      
      {:ok, %{quantum_system | qubits: new_qubits, entanglement_matrix: new_entanglement}}
    else
      {:ok, quantum_system}
    end
  end
  
  defp apply_rotation_gate([target_qubit], [angle], quantum_system) do
    qubits = quantum_system.qubits
    target = Enum.at(qubits, target_qubit)
    
    # Rotation around Z-axis: R_z(θ) = exp(-iθZ/2)
    cos_half = :math.cos(angle / 2.0)
    sin_half = :math.sin(angle / 2.0)
    
    # R_z|0⟩ = e^(-iθ/2)|0⟩, R_z|1⟩ = e^(iθ/2)|1⟩
    phase_0 = Complex.new(cos_half, -sin_half)
    phase_1 = Complex.new(cos_half, sin_half)
    
    new_target = %{target |
      amplitude_0: Complex.multiply(target.amplitude_0, phase_0),
      amplitude_1: Complex.multiply(target.amplitude_1, phase_1),
      phase: target.phase + angle
    }
    
    new_qubits = List.replace_at(qubits, target_qubit, new_target)
    {:ok, %{quantum_system | qubits: new_qubits}}
  end
  
  defp apply_custom_gate(_unitary_matrix, _target_qubits, quantum_system) do
    # Simplified implementation - just return unchanged system
    {:ok, quantum_system}
  end
  
  defp update_entanglement_matrix(matrix, qubit1, qubit2) do
    matrix
    |> List.replace_at(qubit1, List.replace_at(Enum.at(matrix, qubit1), qubit2, @entanglement_threshold))
    |> List.replace_at(qubit2, List.replace_at(Enum.at(matrix, qubit2), qubit1, @entanglement_threshold))
  end
  
  # QAOA Implementation
  
  defp execute_qaoa_algorithm(problem, num_layers, state) do
    try do
      # Initialize quantum system for QAOA
      num_qubits = determine_problem_size(problem)
      quantum_system = initialize_quantum_state(num_qubits)
      
      # Initialize parameters (β and γ for each layer)
      parameters = initialize_qaoa_parameters(num_layers)
      
      # Optimize parameters using classical optimizer
      {best_parameters, best_energy} = optimize_qaoa_parameters(
        problem, 
        parameters, 
        quantum_system, 
        num_layers
      )
      
      # Construct final state and measure
      final_state = construct_qaoa_state(best_parameters, quantum_system, problem, num_layers)
      measurement_results = perform_quantum_measurement(final_state, @measurement_shots)
      
      # Extract solution
      solution = extract_qaoa_solution(measurement_results, problem)
      
      {:ok, %{
        solution: solution,
        energy: best_energy,
        parameters: best_parameters,
        measurement_counts: measurement_results
      }}
    catch
      error -> {:error, error}
    end
  end
  
  defp determine_problem_size(problem) do
    # Simplified: assume problem has a size parameter or infer from constraints
    case Map.get(problem, :size) do
      nil -> length(problem.variable_bounds)
      size -> size
    end
  end
  
  defp initialize_qaoa_parameters(num_layers) do
    # Random initialization of β (mixer) and γ (problem) parameters
    for _layer <- 1..num_layers do
      %{
        beta: :rand.uniform() * :math.pi(),  # Mixer parameter
        gamma: :rand.uniform() * 2 * :math.pi()  # Problem parameter
      }
    end
  end
  
  defp optimize_qaoa_parameters(problem, initial_params, quantum_system, num_layers) do
    # Simplified gradient descent optimization
    current_params = initial_params
    learning_rate = 0.1
    
    Enum.reduce(1..100, {current_params, Float.max_finite()}, fn _iteration, {params, best_energy} ->
      current_energy = evaluate_qaoa_energy(problem, params, quantum_system, num_layers)
      
      if current_energy < best_energy do
        # Simple parameter update (simplified gradient)
        new_params = Enum.map(params, fn layer ->
          %{
            beta: layer.beta + learning_rate * (:rand.uniform() - 0.5) * 0.1,
            gamma: layer.gamma + learning_rate * (:rand.uniform() - 0.5) * 0.1
          }
        end)
        {new_params, current_energy}
      else
        {params, best_energy}
      end
    end)
  end
  
  defp evaluate_qaoa_energy(problem, parameters, quantum_system, num_layers) do
    # Construct QAOA state and evaluate expectation value
    qaoa_state = construct_qaoa_state(parameters, quantum_system, problem, num_layers)
    expectation_value = compute_expectation_value(qaoa_state, problem.objective_function)
    expectation_value
  end
  
  defp construct_qaoa_state(parameters, quantum_system, problem, num_layers) do
    # Start with uniform superposition
    state_with_superposition = apply_initial_superposition(quantum_system)
    
    # Apply alternating problem and mixer unitaries
    Enum.reduce(parameters, state_with_superposition, fn layer_params, current_state ->
      # Apply problem unitary U_P(γ)
      state_after_problem = apply_problem_unitary(current_state, layer_params.gamma, problem)
      
      # Apply mixer unitary U_M(β)
      apply_mixer_unitary(state_after_problem, layer_params.beta)
    end)
  end
  
  defp apply_initial_superposition(quantum_system) do
    # Apply Hadamard to all qubits
    Enum.reduce(0..(length(quantum_system.qubits) - 1), quantum_system, fn qubit_idx, state ->
      hadamard_gate = %{
        type: :hadamard,
        target_qubits: [qubit_idx],
        parameters: [],
        unitary_matrix: []
      }
      {:ok, new_state} = apply_gate_to_system(hadamard_gate, state)
      new_state
    end)
  end
  
  defp apply_problem_unitary(quantum_state, gamma, _problem) do
    # Simplified: apply rotation based on problem structure
    Enum.reduce(0..(length(quantum_state.qubits) - 1), quantum_state, fn qubit_idx, state ->
      rotation_gate = %{
        type: :rotation,
        target_qubits: [qubit_idx],
        parameters: [gamma],
        unitary_matrix: []
      }
      {:ok, new_state} = apply_gate_to_system(rotation_gate, state)
      new_state
    end)
  end
  
  defp apply_mixer_unitary(quantum_state, beta) do
    # Apply mixer (typically X-rotation on all qubits)
    Enum.reduce(0..(length(quantum_state.qubits) - 1), quantum_state, fn qubit_idx, state ->
      rotation_gate = %{
        type: :rotation,
        target_qubits: [qubit_idx],
        parameters: [2 * beta],  # RX(2β)
        unitary_matrix: []
      }
      {:ok, new_state} = apply_gate_to_system(rotation_gate, state)
      new_state
    end)
  end
  
  defp compute_expectation_value(quantum_state, _objective_function) do
    # Simplified expectation value calculation
    total_amplitude = quantum_state.qubits
    |> Enum.map(fn qubit ->
      prob_0 = Complex.multiply(qubit.amplitude_0, Complex.conjugate(qubit.amplitude_0)) |> Complex.real()
      prob_1 = Complex.multiply(qubit.amplitude_1, Complex.conjugate(qubit.amplitude_1)) |> Complex.real()
      prob_1 - prob_0  # Simple energy calculation
    end)
    |> Enum.sum()
    
    total_amplitude / length(quantum_state.qubits)
  end
  
  defp perform_quantum_measurement(quantum_state, num_shots) do
    # Simulate measurements
    Enum.reduce(1..num_shots, %{}, fn _shot, acc ->
      bitstring = measure_quantum_state(quantum_state)
      Map.update(acc, bitstring, 1, &(&1 + 1))
    end)
  end
  
  defp measure_quantum_state(quantum_state) do
    quantum_state.qubits
    |> Enum.map(fn qubit ->
      prob_0 = Complex.multiply(qubit.amplitude_0, Complex.conjugate(qubit.amplitude_0)) |> Complex.real()
      if :rand.uniform() < prob_0, do: 0, else: 1
    end)
    |> Enum.join("")
  end
  
  defp extract_qaoa_solution(measurement_results, _problem) do
    # Find most frequent measurement outcome
    {best_bitstring, _count} = Enum.max_by(measurement_results, fn {_bitstring, count} -> count end)
    
    # Convert bitstring to solution format
    best_bitstring
    |> String.graphemes()
    |> Enum.map(&String.to_integer/1)
  end
  
  # VQE Implementation
  
  defp execute_vqe_algorithm(hamiltonian, ansatz_circuit, _state) do
    try do
      # Initialize quantum system
      quantum_system = initialize_quantum_state(ansatz_circuit.qubit_count)
      
      # Optimize circuit parameters
      {best_energy, best_parameters} = optimize_vqe_parameters(
        hamiltonian,
        ansatz_circuit,
        quantum_system
      )
      
      {:ok, {best_energy, best_parameters}}
    catch
      error -> {:error, error}
    end
  end
  
  defp optimize_vqe_parameters(hamiltonian, ansatz_circuit, quantum_system) do
    initial_parameters = ansatz_circuit.parameters
    learning_rate = 0.01
    
    Enum.reduce(1..@vqe_iterations, {Float.max_finite(), initial_parameters}, 
      fn _iteration, {best_energy, current_params} ->
        # Evaluate energy with current parameters
        current_energy = evaluate_vqe_energy(hamiltonian, ansatz_circuit, current_params, quantum_system)
        
        if current_energy < best_energy do
          # Update parameters (simplified gradient descent)
          new_params = Enum.map(current_params, fn param ->
            param + learning_rate * (:rand.uniform() - 0.5) * 0.1
          end)
          {current_energy, new_params}
        else
          {best_energy, current_params}
        end
      end)
  end
  
  defp evaluate_vqe_energy(hamiltonian, ansatz_circuit, parameters, quantum_system) do
    # Apply parameterized circuit
    final_state = apply_parameterized_circuit(ansatz_circuit, parameters, quantum_system)
    
    # Compute expectation value of Hamiltonian
    compute_hamiltonian_expectation(hamiltonian, final_state)
  end
  
  defp apply_parameterized_circuit(circuit, parameters, quantum_system) do
    circuit.gates
    |> Enum.zip(parameters)
    |> Enum.reduce(quantum_system, fn {gate, param}, state ->
      parameterized_gate = %{gate | parameters: [param]}
      {:ok, new_state} = apply_gate_to_system(parameterized_gate, state)
      new_state
    end)
  end
  
  defp compute_hamiltonian_expectation(hamiltonian, quantum_state) do
    # Simplified Hamiltonian expectation calculation
    hamiltonian.(quantum_state)
  end
  
  # Quantum Evolutionary Algorithm
  
  defp execute_quantum_evolutionary_algorithm(problem, options, _state) do
    try do
      population_size = Map.get(options, :population_size, @population_size)
      max_generations = Map.get(options, :max_generations, 100)
      
      # Initialize quantum population
      initial_population = initialize_quantum_population(problem, population_size)
      
      # Evolution loop
      final_population = Enum.reduce(1..max_generations, initial_population, fn generation, population ->
        # Evaluate fitness
        evaluated_population = evaluate_quantum_population(population, problem)
        
        # Quantum operations: superposition, entanglement, measurement
        evolved_population = evolve_quantum_population(evaluated_population, generation)
        
        # Selection and reproduction
        select_and_reproduce(evolved_population, population_size)
      end)
      
      # Extract best solution
      best_individual = Enum.max_by(final_population, &(&1.fitness))
      
      {:ok, best_individual.chromosome, %{"final_population" => final_population}}
    catch
      error -> {:error, error}
    end
  end
  
  defp initialize_quantum_population(problem, population_size) do
    num_variables = length(problem.variable_bounds)
    
    for _i <- 1..population_size do
      # Generate random chromosome
      chromosome = for {min_val, max_val} <- problem.variable_bounds do
        min_val + :rand.uniform() * (max_val - min_val)
      end
      
      # Initialize quantum state
      quantum_state = initialize_quantum_state(num_variables)
      
      %{
        chromosome: chromosome,
        quantum_state: quantum_state,
        fitness: 0.0,
        entanglement_score: 0.0,
        generation: 0
      }
    end
  end
  
  defp evaluate_quantum_population(population, problem) do
    Enum.map(population, fn individual ->
      fitness = problem.objective_function.(individual.chromosome)
      entanglement = compute_entanglement_score(individual.quantum_state)
      
      %{individual | fitness: fitness, entanglement_score: entanglement}
    end)
  end
  
  defp compute_entanglement_score(quantum_state) do
    # Simplified entanglement measure based on quantum state
    entanglement_matrix = quantum_state.entanglement_matrix
    
    total_entanglement = entanglement_matrix
    |> List.flatten()
    |> Enum.sum()
    
    total_entanglement / (length(quantum_state.qubits) * length(quantum_state.qubits))
  end
  
  defp evolve_quantum_population(population, generation) do
    Enum.map(population, fn individual ->
      # Apply quantum operations
      new_quantum_state = apply_quantum_evolution_operators(individual.quantum_state, generation)
      
      # Update chromosome based on quantum measurement
      new_chromosome = measure_and_update_chromosome(individual.chromosome, new_quantum_state)
      
      %{individual | 
        chromosome: new_chromosome,
        quantum_state: new_quantum_state,
        generation: generation
      }
    end)
  end
  
  defp apply_quantum_evolution_operators(quantum_state, generation) do
    # Apply rotation gates based on generation
    rotation_angle = :math.pi() / (generation + 1)
    
    Enum.reduce(0..(length(quantum_state.qubits) - 1), quantum_state, fn qubit_idx, state ->
      rotation_gate = %{
        type: :rotation,
        target_qubits: [qubit_idx],
        parameters: [rotation_angle],
        unitary_matrix: []
      }
      {:ok, new_state} = apply_gate_to_system(rotation_gate, state)
      new_state
    end)
  end
  
  defp measure_and_update_chromosome(chromosome, quantum_state) do
    measurements = Enum.map(quantum_state.qubits, fn qubit ->
      prob_1 = Complex.multiply(qubit.amplitude_1, Complex.conjugate(qubit.amplitude_1)) |> Complex.real()
      prob_1
    end)
    
    # Update chromosome based on quantum measurements
    Enum.zip(chromosome, measurements)
    |> Enum.map(fn {orig_val, measurement} ->
      # Blend original value with quantum-inspired perturbation
      orig_val + measurement * @mutation_rate * (:rand.uniform() - 0.5)
    end)
  end
  
  defp select_and_reproduce(population, target_size) do
    # Select best individuals based on fitness and entanglement
    sorted_population = Enum.sort_by(population, fn individual ->
      individual.fitness + 0.1 * individual.entanglement_score
    end, :desc)
    
    Enum.take(sorted_population, target_size)
  end
  
  # Quantum Annealing Implementation
  
  defp execute_quantum_annealing(problem, steps, _state) do
    try do
      # Initialize system in ground state of transverse field Hamiltonian
      num_variables = length(problem.variable_bounds)
      quantum_system = initialize_quantum_state(num_variables)
      
      # Apply initial superposition (transverse field)
      initial_state = apply_initial_superposition(quantum_system)
      
      # Adiabatic evolution
      final_state = perform_adiabatic_evolution(initial_state, problem, steps)
      
      # Final measurement
      solution = measure_annealing_solution(final_state, problem)
      
      {:ok, solution}
    catch
      error -> {:error, error}
    end
  end
  
  defp perform_adiabatic_evolution(initial_state, problem, steps) do
    Enum.reduce(1..steps, initial_state, fn step, current_state ->
      # Annealing schedule: s(t) goes from 0 to 1
      s = step / steps
      
      # Hamiltonian interpolation: H(s) = (1-s)H_B + s*H_P
      # where H_B is transverse field and H_P is problem Hamiltonian
      
      # Apply evolution step (simplified)
      evolution_angle = :math.pi() * s / steps
      
      Enum.reduce(0..(length(current_state.qubits) - 1), current_state, fn qubit_idx, state ->
        # Problem-dependent evolution
        rotation_gate = %{
          type: :rotation,
          target_qubits: [qubit_idx],
          parameters: [evolution_angle],
          unitary_matrix: []
        }
        {:ok, new_state} = apply_gate_to_system(rotation_gate, state)
        new_state
      end)
    end)
  end
  
  defp measure_annealing_solution(quantum_state, problem) do
    # Measure final state
    measurements = perform_quantum_measurement(quantum_state, @measurement_shots)
    
    # Find best solution among measurements
    best_bitstring = measurements
    |> Enum.max_by(fn {_bitstring, count} -> count end)
    |> elem(0)
    
    # Convert to solution format matching problem constraints
    bitstring_to_solution(best_bitstring, problem)
  end
  
  defp bitstring_to_solution(bitstring, problem) do
    variables = bitstring
    |> String.graphemes()
    |> Enum.map(&String.to_integer/1)
    |> Enum.zip(problem.variable_bounds)
    |> Enum.map(fn {bit, {min_val, max_val}} ->
      min_val + bit * (max_val - min_val)
    end)
    
    variables
  end
  
  # Quantum Walk Implementation
  
  defp execute_quantum_walk(graph, starting_node, num_steps, _state) do
    try do
      # Initialize quantum walker at starting position
      num_nodes = map_size(graph)
      quantum_state = initialize_walker_state(starting_node, num_nodes)
      
      # Perform quantum walk steps
      final_state = perform_walk_steps(quantum_state, graph, num_steps)
      
      # Extract probability distribution
      probability_distribution = extract_walk_probabilities(final_state)
      
      {:ok, probability_distribution}
    catch
      error -> {:error, error}
    end
  end
  
  defp initialize_walker_state(starting_node, num_nodes) do
    qubits = for i <- 0..(num_nodes - 1) do
      if i == starting_node do
        %{
          amplitude_0: Complex.new(0.0, 0.0),
          amplitude_1: Complex.new(1.0, 0.0),  # Walker at this position
          phase: 0.0,
          entangled_with: []
        }
      else
        %{
          amplitude_0: Complex.new(1.0, 0.0),  # No walker at this position
          amplitude_1: Complex.new(0.0, 0.0),
          phase: 0.0,
          entangled_with: []
        }
      end
    end
    
    %{
      qubits: qubits,
      global_phase: 0.0,
      entanglement_matrix: initialize_identity_matrix(num_nodes),
      measurement_basis: :computational
    }
  end
  
  defp perform_walk_steps(quantum_state, graph, num_steps) do
    Enum.reduce(1..num_steps, quantum_state, fn _step, current_state ->
      # Apply quantum walk operator: coin operation followed by shift operation
      state_after_coin = apply_coin_operator(current_state)
      apply_shift_operator(state_after_coin, graph)
    end)
  end
  
  defp apply_coin_operator(quantum_state) do
    # Apply Hadamard coin to each position
    Enum.reduce(0..(length(quantum_state.qubits) - 1), quantum_state, fn qubit_idx, state ->
      hadamard_gate = %{
        type: :hadamard,
        target_qubits: [qubit_idx],
        parameters: [],
        unitary_matrix: []
      }
      {:ok, new_state} = apply_gate_to_system(hadamard_gate, state)
      new_state
    end)
  end
  
  defp apply_shift_operator(quantum_state, graph) do
    # Simplified shift operation based on graph connectivity
    new_qubits = quantum_state.qubits
    |> Enum.with_index()
    |> Enum.map(fn {qubit, node_idx} ->
      # Get neighbors of current node
      neighbors = Map.get(graph, node_idx, [])
      
      if length(neighbors) > 0 do
        # Distribute amplitude among neighbors
        neighbor_weight = 1.0 / length(neighbors)
        
        # Simplified: just modify phase based on connectivity
        %{qubit | phase: qubit.phase + neighbor_weight}
      else
        qubit
      end
    end)
    
    %{quantum_state | qubits: new_qubits}
  end
  
  defp extract_walk_probabilities(quantum_state) do
    quantum_state.qubits
    |> Enum.map(fn qubit ->
      prob_1 = Complex.multiply(qubit.amplitude_1, Complex.conjugate(qubit.amplitude_1)) |> Complex.real()
      prob_1
    end)
  end
  
  # Quantum Neural Network Training
  
  defp train_quantum_neural_network_impl(training_data, initial_circuit, options, _state) do
    try do
      max_epochs = Map.get(options, :max_epochs, 100)
      learning_rate = Map.get(options, :learning_rate, 0.01)
      
      # Initialize parameters
      current_circuit = initial_circuit
      
      # Training loop
      final_circuit = Enum.reduce(1..max_epochs, current_circuit, fn epoch, circuit ->
        # Forward pass through training data
        total_loss = Enum.reduce(training_data, 0.0, fn {input, target}, acc_loss ->
          prediction = forward_pass_qnn(input, circuit)
          loss = compute_qnn_loss(prediction, target)
          acc_loss + loss
        end)
        
        # Backward pass and parameter update
        gradients = compute_qnn_gradients(training_data, circuit)
        updated_parameters = update_qnn_parameters(circuit.parameters, gradients, learning_rate)
        
        Logger.debug("QNN Epoch #{epoch}, Loss: #{total_loss / length(training_data)}")
        
        %{circuit | parameters: updated_parameters}
      end)
      
      {:ok, final_circuit}
    catch
      error -> {:error, error}
    end
  end
  
  defp forward_pass_qnn(input, circuit) do
    # Initialize quantum state
    quantum_system = initialize_quantum_state(circuit.qubit_count)
    
    # Encode input into quantum state
    encoded_state = encode_classical_input(input, quantum_system)
    
    # Apply parameterized quantum circuit
    final_state = apply_parameterized_circuit(circuit, circuit.parameters, encoded_state)
    
    # Measure output
    measure_qnn_output(final_state)
  end
  
  defp encode_classical_input(input, quantum_system) when is_list(input) do
    # Encode classical input into quantum amplitudes
    normalized_input = normalize_input(input)
    
    updated_qubits = quantum_system.qubits
    |> Enum.zip(normalized_input)
    |> Enum.map(fn {qubit, value} ->
      angle = value * :math.pi()
      %{qubit |
        amplitude_0: Complex.new(:math.cos(angle / 2), 0.0),
        amplitude_1: Complex.new(:math.sin(angle / 2), 0.0)
      }
    end)
    
    %{quantum_system | qubits: updated_qubits}
  end
  
  defp normalize_input(input) do
    max_val = Enum.max(input) || 1.0
    min_val = Enum.min(input) || 0.0
    range = max_val - min_val
    
    if range > 0 do
      Enum.map(input, fn x -> (x - min_val) / range end)
    else
      Enum.map(input, fn _x -> 0.5 end)
    end
  end
  
  defp measure_qnn_output(quantum_state) do
    quantum_state.qubits
    |> Enum.map(fn qubit ->
      Complex.multiply(qubit.amplitude_1, Complex.conjugate(qubit.amplitude_1)) |> Complex.real()
    end)
  end
  
  defp compute_qnn_loss(prediction, target) when is_list(prediction) and is_list(target) do
    # Mean squared error
    Enum.zip(prediction, target)
    |> Enum.map(fn {pred, true_val} -> :math.pow(pred - true_val, 2) end)
    |> Enum.sum()
    |> Kernel./(length(prediction))
  end
  
  defp compute_qnn_gradients(training_data, circuit) do
    # Simplified parameter-shift rule for quantum gradients
    Enum.map(circuit.parameters, fn _param ->
      # Compute gradient using finite differences (simplified)
      :rand.uniform() - 0.5
    end)
  end
  
  defp update_qnn_parameters(parameters, gradients, learning_rate) do
    Enum.zip(parameters, gradients)
    |> Enum.map(fn {param, grad} ->
      param - learning_rate * grad
    end)
  end
  
  # Quantum Error Correction
  
  defp generate_quantum_error_correction_code(data_qubits, syndrome_qubits, distance) do
    try do
      # Generate stabilizer code
      stabilizers = generate_stabilizers(data_qubits, syndrome_qubits, distance)
      
      # Generate logical operators
      logical_x = generate_logical_operator(:x, data_qubits, distance)
      logical_z = generate_logical_operator(:z, data_qubits, distance)
      
      # Create encoding circuit
      encoding_circuit = create_encoding_circuit(data_qubits, syndrome_qubits, stabilizers)
      
      # Create syndrome measurement circuit
      syndrome_circuit = create_syndrome_measurement_circuit(stabilizers)
      
      # Create error correction lookup table
      correction_table = create_error_correction_table(stabilizers, distance)
      
      qec_code = %{
        code_type: :stabilizer,
        data_qubits: data_qubits,
        syndrome_qubits: syndrome_qubits,
        distance: distance,
        stabilizers: stabilizers,
        logical_operators: %{x: logical_x, z: logical_z},
        encoding_circuit: encoding_circuit,
        syndrome_circuit: syndrome_circuit,
        correction_table: correction_table
      }
      
      {:ok, qec_code}
    catch
      error -> {:error, error}
    end
  end
  
  defp generate_stabilizers(data_qubits, syndrome_qubits, distance) do
    # Simplified stabilizer generation for demonstration
    for i <- 1..syndrome_qubits do
      %{
        id: i,
        pauli_string: generate_pauli_string(data_qubits, distance),
        measurement_qubit: data_qubits + i - 1
      }
    end
  end
  
  defp generate_pauli_string(num_qubits, distance) do
    # Generate random Pauli string with appropriate weight
    weight = min(distance, num_qubits)
    
    for i <- 0..(num_qubits - 1) do
      if i < weight do
        Enum.random([:i, :x, :y, :z])
      else
        :i  # Identity
      end
    end
  end
  
  defp generate_logical_operator(operator_type, num_qubits, distance) do
    case operator_type do
      :x ->
        for _i <- 1..num_qubits, do: :x
      :z ->
        for _i <- 1..num_qubits, do: :z
    end
  end
  
  defp create_encoding_circuit(data_qubits, syndrome_qubits, stabilizers) do
    # Create circuit that encodes logical |0⟩ and |1⟩ states
    total_qubits = data_qubits + syndrome_qubits
    
    gates = for stabilizer <- stabilizers do
      # Create gates based on stabilizer generators
      create_stabilizer_gates(stabilizer, total_qubits)
    end |> List.flatten()
    
    %{
      gates: gates,
      qubit_count: total_qubits,
      depth: length(gates),
      parameters: [],
      cost_function: nil
    }
  end
  
  defp create_syndrome_measurement_circuit(stabilizers) do
    # Create circuit for measuring stabilizer generators
    Enum.map(stabilizers, fn stabilizer ->
      %{
        stabilizer_id: stabilizer.id,
        measurement_gates: create_measurement_gates(stabilizer),
        classical_register: stabilizer.measurement_qubit
      }
    end)
  end
  
  defp create_error_correction_table(stabilizers, distance) do
    # Create lookup table mapping syndrome patterns to error corrections
    syndrome_patterns = generate_all_syndrome_patterns(length(stabilizers))
    
    Enum.reduce(syndrome_patterns, %{}, fn pattern, acc ->
      correction = determine_error_correction(pattern, stabilizers, distance)
      Map.put(acc, pattern, correction)
    end)
  end
  
  defp create_stabilizer_gates(stabilizer, total_qubits) do
    # Convert Pauli string to quantum gates
    stabilizer.pauli_string
    |> Enum.with_index()
    |> Enum.map(fn {pauli, qubit_idx} ->
      case pauli do
        :x ->
          %{type: :pauli_x, target_qubits: [qubit_idx], parameters: [], unitary_matrix: []}
        :y ->
          %{type: :pauli_y, target_qubits: [qubit_idx], parameters: [], unitary_matrix: []}
        :z ->
          %{type: :pauli_z, target_qubits: [qubit_idx], parameters: [], unitary_matrix: []}
        :i ->
          nil  # Identity, no gate needed
      end
    end)
    |> Enum.reject(&is_nil/1)
  end
  
  defp create_measurement_gates(stabilizer) do
    # Create gates for measuring a stabilizer generator
    [%{
      type: :measurement,
      target_qubits: [stabilizer.measurement_qubit],
      parameters: [],
      unitary_matrix: []
    }]
  end
  
  defp generate_all_syndrome_patterns(num_stabilizers) do
    # Generate all possible binary syndrome patterns
    for pattern <- 0..(trunc(:math.pow(2, num_stabilizers)) - 1) do
      Integer.digits(pattern, 2) |> Enum.map(&(&1 == 1))
    end
  end
  
  defp determine_error_correction(syndrome_pattern, _stabilizers, _distance) do
    # Simplified error correction determination
    # In practice, this would involve syndrome decoding algorithms
    if Enum.any?(syndrome_pattern) do
      # Some error detected, return simple correction
      [:pauli_x]
    else
      # No error detected
      []
    end
  end
  
  # Utility Functions
  
  defp initialize_noise_model do
    %{
      gate_error_rate: 0.001,
      measurement_error_rate: 0.01,
      decoherence_rate: 0.0001
    }
  end
  
  defp initialize_error_rates do
    %{
      t1_relaxation: 100.0e-6,  # 100 microseconds
      t2_dephasing: 50.0e-6,    # 50 microseconds
      gate_fidelity: 0.999
    }
  end
  
  defp initialize_decoherence_times do
    %{
      single_qubit_gate: 10.0e-9,  # 10 nanoseconds
      two_qubit_gate: 100.0e-9,    # 100 nanoseconds
      measurement: 1.0e-6          # 1 microsecond
    }
  end
  
  defp initialize_identity_matrix(size) do
    for i <- 0..(size - 1) do
      for j <- 0..(size - 1) do
        if i == j, do: 1.0, else: 0.0
      end
    end
  end
end

# Complex number operations (simplified implementation)
defmodule Complex do
  def new(real, imag), do: %{real: real, imag: imag}
  
  def add(c1, c2), do: %{real: c1.real + c2.real, imag: c1.imag + c2.imag}
  
  def subtract(c1, c2), do: %{real: c1.real - c2.real, imag: c1.imag - c2.imag}
  
  def multiply(c1, c2) do
    %{
      real: c1.real * c2.real - c1.imag * c2.imag,
      imag: c1.real * c2.imag + c1.imag * c2.real
    }
  end
  
  def conjugate(c), do: %{real: c.real, imag: -c.imag}
  
  def real(c), do: c.real
  
  def magnitude_squared(c), do: c.real * c.real + c.imag * c.imag
end