Inference Engines¶
This document describes the inference algorithms and engines that power probabilistic reasoning in quantum-classical hybrid networks.
Overview¶
The inference system combines classical probabilistic reasoning with quantum amplitude-based computation to provide powerful uncertainty modeling capabilities:
graph TD
A[Inference Engine] --> B[Belief Propagation]
A --> C[Variational Methods]
A --> D[Causal Inference]
A --> E[Quantum Algorithms]
B --> F[Message Passing]
B --> G[Amplitude Propagation]
C --> H[VQE]
C --> I[QAOA]
D --> J[Do-Calculus]
D --> K[Counterfactuals]
E --> L[Grover Search]
E --> M[Quantum Sampling]Inference Engine Architecture¶
The main inference coordinator manages different reasoning algorithms:
from probabilistic_quantum_reasoner.inference import InferenceEngine
class InferenceEngine:
"""Main inference engine for quantum Bayesian networks."""
def __init__(self, network: QuantumBayesianNetwork):
self.network = network
self.belief_propagation = QuantumBeliefPropagation(network)
self.variational_engine = VariationalInference(network)
self.causal_engine = CausalInference(network)
def infer(self, query_nodes: List[str], evidence: Dict[str, Any] = None,
method: str = "belief_propagation", **kwargs) -> InferenceResult:
"""Perform probabilistic inference."""
if method == "belief_propagation":
return self.belief_propagation.infer(query_nodes, evidence, **kwargs)
elif method == "variational":
return self.variational_engine.infer(query_nodes, evidence, **kwargs)
elif method == "sampling":
return self._quantum_sampling(query_nodes, evidence, **kwargs)
else:
raise ValueError(f"Unknown inference method: {method}")
Belief Propagation¶
Quantum belief propagation extends classical message passing to quantum amplitude domains.
Quantum Message Passing¶
class QuantumBeliefPropagation:
"""Quantum belief propagation algorithm."""
def propagate_messages(self, max_iterations: int = 50) -> bool:
"""Propagate quantum amplitude messages through network."""
for iteration in range(max_iterations):
converged = True
for edge in self.network.edges:
# Compute quantum message from parent to child
old_message = self.messages.get(edge, None)
new_message = self._compute_quantum_message(edge)
# Check convergence
if old_message is not None:
fidelity = self._compute_message_fidelity(old_message, new_message)
if fidelity < 0.999: # Convergence threshold
converged = False
self.messages[edge] = new_message
if converged:
return True
return False # Did not converge
def _compute_quantum_message(self, edge: Tuple[str, str]) -> QuantumMessage:
"""Compute quantum amplitude message along edge."""
parent_id, child_id = edge
parent_node = self.network.nodes[parent_id]
child_node = self.network.nodes[child_id]
if isinstance(parent_node, QuantumNode):
# Quantum parent sends amplitude message
parent_amplitudes = parent_node.quantum_state.amplitudes
# Apply conditional probability transformation
if hasattr(child_node, 'conditional_probability_table'):
cpt = child_node.conditional_probability_table
message_amplitudes = self._apply_cpt_to_amplitudes(
parent_amplitudes, cpt
)
else:
# Direct amplitude propagation
message_amplitudes = parent_amplitudes
return QuantumMessage(
amplitudes=message_amplitudes,
sender=parent_id,
receiver=child_id
)
else:
# Classical parent sends probability message (converted to amplitudes)
parent_probs = parent_node.probability_distribution
message_amplitudes = np.sqrt(list(parent_probs.values())).astype(complex)
return QuantumMessage(
amplitudes=message_amplitudes,
sender=parent_id,
receiver=child_id
)
Amplitude Integration¶
def _integrate_messages(self, node_id: str) -> QuantumState:
"""Integrate incoming quantum messages at a node."""
node = self.network.nodes[node_id]
incoming_messages = [msg for msg in self.messages.values()
if msg.receiver == node_id]
if isinstance(node, QuantumNode):
# Start with node's intrinsic quantum state
combined_amplitudes = node.quantum_state.amplitudes.copy()
# Multiply by incoming amplitude messages
for message in incoming_messages:
combined_amplitudes = self._multiply_amplitudes(
combined_amplitudes, message.amplitudes
)
# Normalize
combined_amplitudes /= np.linalg.norm(combined_amplitudes)
return QuantumState(combined_amplitudes)
else:
# Classical node: convert messages to probabilities
combined_probs = np.ones(len(node.outcome_space))
for message in incoming_messages:
message_probs = np.abs(message.amplitudes) ** 2
combined_probs *= message_probs
# Normalize and convert back to amplitudes
combined_probs /= np.sum(combined_probs)
combined_amplitudes = np.sqrt(combined_probs).astype(complex)
return QuantumState(combined_amplitudes)
Variational Inference¶
Variational methods optimize parameterized quantum circuits to approximate posterior distributions.
Variational Quantum Eigensolver (VQE)¶
class VariationalInference:
"""Variational quantum inference using VQE."""
def __init__(self, network: QuantumBayesianNetwork, backend: QuantumBackend):
self.network = network
self.backend = backend
def optimize_variational_distribution(self, target_nodes: List[str],
evidence: Dict[str, Any] = None,
max_iterations: int = 100) -> Dict[str, Any]:
"""Optimize variational quantum circuit to approximate posterior."""
# Create parameterized quantum circuit
n_qubits = len(target_nodes)
n_parameters = 2 * n_qubits # RY and RZ rotations per qubit
def create_ansatz(parameters: np.ndarray):
"""Create variational ansatz circuit."""
circuit_ops = []
# Layer 1: Individual rotations
for i, param in enumerate(parameters[:n_qubits]):
circuit_ops.append(('RY', i, param))
# Layer 2: Entangling gates
for i in range(n_qubits - 1):
circuit_ops.append(('CNOT', i, i + 1))
# Layer 3: More rotations
for i, param in enumerate(parameters[n_qubits:2*n_qubits]):
circuit_ops.append(('RZ', i, param))
return circuit_ops
# Cost function: KL divergence from target distribution
def cost_function(parameters: np.ndarray) -> float:
circuit = create_ansatz(parameters)
variational_state = self._execute_circuit(circuit)
# Compute target distribution from network inference
target_result = self.network.infer(
query_nodes=target_nodes,
evidence=evidence,
method="belief_propagation"
)
# Extract target probabilities
target_probs = []
for node_id in target_nodes:
node_dist = target_result.marginal_probabilities[node_id]
target_probs.extend(list(node_dist.values()))
target_probs = np.array(target_probs)
variational_probs = np.abs(variational_state.amplitudes) ** 2
# KL divergence
epsilon = 1e-10
kl_div = np.sum(target_probs * np.log(
(target_probs + epsilon) / (variational_probs + epsilon)
))
return kl_div
# Optimization using classical optimizer
from scipy.optimize import minimize
initial_params = np.random.uniform(0, 2*np.pi, n_parameters)
result = minimize(
cost_function,
initial_params,
method='L-BFGS-B',
options={'maxiter': max_iterations}
)
# Return optimized parameters and final state
optimal_circuit = create_ansatz(result.x)
optimal_state = self._execute_circuit(optimal_circuit)
return {
"optimal_parameters": result.x,
"optimal_state": optimal_state,
"final_cost": result.fun,
"optimization_success": result.success,
"iterations": result.nit
}
Quantum Approximate Optimization Algorithm (QAOA)¶
def qaoa_inference(self, query_nodes: List[str], evidence: Dict[str, Any] = None,
p_layers: int = 3) -> Dict[str, Any]:
"""Use QAOA for approximate inference."""
# Define problem Hamiltonian based on network structure
problem_hamiltonian = self._construct_problem_hamiltonian(query_nodes, evidence)
# Define mixer Hamiltonian (usually sum of X rotations)
mixer_hamiltonian = self._construct_mixer_hamiltonian(len(query_nodes))
# QAOA ansatz with p layers
def qaoa_circuit(gamma_params: np.ndarray, beta_params: np.ndarray):
"""QAOA circuit with alternating problem and mixer unitaries."""
# Initialize in uniform superposition
state = np.ones(2**len(query_nodes), dtype=complex) / np.sqrt(2**len(query_nodes))
for p in range(p_layers):
# Apply problem unitary: exp(-i * gamma * H_problem)
problem_unitary = self._matrix_exponential(-1j * gamma_params[p] * problem_hamiltonian)
state = problem_unitary @ state
# Apply mixer unitary: exp(-i * beta * H_mixer)
mixer_unitary = self._matrix_exponential(-1j * beta_params[p] * mixer_hamiltonian)
state = mixer_unitary @ state
return state
# Optimize QAOA parameters
def qaoa_cost(params):
gamma_params = params[:p_layers]
beta_params = params[p_layers:]
final_state = qaoa_circuit(gamma_params, beta_params)
expectation = np.real(np.conj(final_state) @ problem_hamiltonian @ final_state)
return expectation
# Classical optimization
initial_params = np.random.uniform(0, np.pi, 2 * p_layers)
result = minimize(qaoa_cost, initial_params, method='COBYLA')
# Extract final quantum state
optimal_gamma = result.x[:p_layers]
optimal_beta = result.x[p_layers:]
final_state = qaoa_circuit(optimal_gamma, optimal_beta)
return {
"optimal_state": final_state,
"optimal_parameters": {"gamma": optimal_gamma, "beta": optimal_beta},
"expectation_value": result.fun,
"optimization_success": result.success
}
Causal Inference¶
Implement do-calculus and counterfactual reasoning in quantum networks.
Do-Calculus Implementation¶
class CausalInference:
"""Causal inference with quantum interventions."""
def do_intervention(self, interventions: Dict[str, Any],
query_nodes: List[str]) -> InferenceResult:
"""Perform causal intervention using do-calculus."""
# Create modified network with interventions
modified_network = self.network.copy()
for node_id, intervention_value in interventions.items():
node = modified_network.nodes[node_id]
if isinstance(node, QuantumNode):
# Quantum intervention: set node to specific amplitude pattern
intervention_amplitudes = self._create_intervention_amplitudes(
node.outcome_space, intervention_value
)
node.set_amplitudes(intervention_amplitudes)
# Remove incoming edges (break causal dependencies)
incoming_edges = [edge for edge in modified_network.edges
if edge[1] == node_id]
for edge in incoming_edges:
modified_network.remove_edge(edge)
else:
# Classical intervention: set deterministic distribution
intervention_dist = {value: 1.0 if value == intervention_value else 0.0
for value in node.outcome_space}
node.set_distribution(intervention_dist)
# Remove incoming edges
incoming_edges = [edge for edge in modified_network.edges
if edge[1] == node_id]
for edge in incoming_edges:
modified_network.remove_edge(edge)
# Perform inference on modified network
return modified_network.infer(query_nodes=query_nodes)
Counterfactual Reasoning¶
def counterfactual_inference(self, factual_evidence: Dict[str, Any],
counterfactual_intervention: Dict[str, Any],
query_nodes: List[str]) -> Dict[str, Any]:
"""Perform counterfactual reasoning: "What if X had been Y instead of Z?"."""
# Step 1: Abduction - infer latent variables given factual evidence
latent_inference = self.network.infer(
query_nodes=self._get_latent_nodes(),
evidence=factual_evidence
)
# Step 2: Action - apply counterfactual intervention
counterfactual_network = self.network.copy()
# Set latent variables to inferred values
for latent_id in self._get_latent_nodes():
latent_dist = latent_inference.marginal_probabilities[latent_id]
# Sample from posterior or use expected value
latent_value = max(latent_dist.items(), key=lambda x: x[1])[0]
counterfactual_network.set_evidence(latent_id, latent_value)
# Apply intervention
counterfactual_result = counterfactual_network.intervene(
interventions=counterfactual_intervention,
query_nodes=query_nodes
)
# Step 3: Prediction - compute counterfactual outcome
return {
"factual_evidence": factual_evidence,
"counterfactual_intervention": counterfactual_intervention,
"latent_posterior": latent_inference.marginal_probabilities,
"counterfactual_outcome": counterfactual_result.marginal_probabilities,
"causal_effect": self._compute_causal_effect(
factual_evidence, counterfactual_result.marginal_probabilities
)
}
Quantum-Specific Algorithms¶
Grover-Based Search¶
def grover_inference(self, query_predicate: callable,
max_iterations: Optional[int] = None) -> Dict[str, Any]:
"""Use Grover's algorithm for quantum search-based inference."""
n_qubits = len(self.network.quantum_nodes)
n_items = 2 ** n_qubits
if max_iterations is None:
max_iterations = int(np.pi * np.sqrt(n_items) / 4)
# Initialize uniform superposition
state = np.ones(n_items, dtype=complex) / np.sqrt(n_items)
# Grover iterations
for iteration in range(max_iterations):
# Oracle: mark satisfying states
oracle_matrix = self._construct_oracle_matrix(query_predicate)
state = oracle_matrix @ state
# Diffusion operator: amplify marked amplitudes
diffusion_matrix = self._construct_diffusion_matrix(n_items)
state = diffusion_matrix @ state
# Measure final state
probabilities = np.abs(state) ** 2
return {
"amplitudes": state,
"probabilities": probabilities,
"most_likely_state": np.argmax(probabilities),
"iterations": max_iterations
}
Performance Characteristics¶
Complexity Analysis¶
| Algorithm | Time Complexity | Space Complexity | Quantum Advantage |
|---|---|---|---|
| Belief Propagation | O(n²d^k) | O(nd) | Amplitude interference |
| Variational | O(poly(n)) | O(2^n) | Circuit expressivity |
| QAOA | O(p·poly(n)) | O(2^n) | Quantum parallelism |
| Grover Search | O(√N) | O(log N) | Quadratic speedup |
Where: - n = number of nodes - d = domain size - k = maximum node degree - p = QAOA layers - N = search space size
Accuracy vs Speed Trade-offs¶
# Configuration for different use cases
INFERENCE_CONFIGS = {
"fast": {
"method": "belief_propagation",
"max_iterations": 10,
"convergence_threshold": 1e-3
},
"accurate": {
"method": "variational",
"max_iterations": 1000,
"convergence_threshold": 1e-6
},
"quantum_advantage": {
"method": "grover",
"max_iterations": None, # Optimal
"post_processing": True
}
}
Next Steps¶
- Learn about building networks
- Explore causal reasoning in detail
- See variational methods examples
- Check API Reference for complete documentation