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"""
High-Fidelity Quantum Approximate Optimization Algorithm (QAOA) Simulation using Google Cirq.
QAOA is a hybrid quantum-classical algorithm designed for solving combinatorial optimization problems.
This implementation demonstrates QAOA for solving the MaxCut problem on small graphs.
"""
import cirq
import numpy as np
import networkx as nx
import sympy
import random
from cirq.contrib.svg import circuit_to_svg
def create_maxcut_cost_operator(qubits, graph):
"""
Creates the cost Hamiltonian for the MaxCut problem.
Args:
qubits: Dictionary mapping node indices to qubits
graph: NetworkX graph representing the problem
Returns:
A cirq.Circuit implementing the cost Hamiltonian evolution
"""
circuit = cirq.Circuit()
# The cost Hamiltonian for MaxCut is sum_(i,j)∈E (1 - Z_i Z_j)/2
# For circuit implementation, we use ZZ rotations
for i, j in graph.edges():
circuit.append(cirq.ZZ(qubits[i], qubits[j])**sympy.Symbol('gamma'))
return circuit
def create_mixer_operator(qubits):
"""
Creates the mixer Hamiltonian for QAOA.
Args:
qubits: List of qubits
Returns:
A cirq.Circuit implementing the mixer Hamiltonian evolution
"""
circuit = cirq.Circuit()
# The mixer Hamiltonian is sum_i X_i
# For circuit implementation, we use X rotations
for qubit in qubits.values():
circuit.append(cirq.X(qubit)**sympy.Symbol('beta'))
return circuit
def create_qaoa_circuit(qubits, graph, p=1):
"""
Creates a QAOA circuit for the MaxCut problem.
Args:
qubits: Dictionary mapping node indices to qubits
graph: NetworkX graph representing the problem
p: Number of QAOA layers
Returns:
A cirq.Circuit implementing QAOA
"""
circuit = cirq.Circuit()
# Initialize in superposition
circuit.append(cirq.H.on_each(*qubits.values()))
# Apply p layers of QAOA
for layer in range(p):
# Apply cost Hamiltonian with parameter gamma_layer
gamma = sympy.Symbol(f'gamma_{layer}')
for i, j in graph.edges():
circuit.append(cirq.ZZ(qubits[i], qubits[j])**gamma)
# Apply mixer Hamiltonian with parameter beta_layer
beta = sympy.Symbol(f'beta_{layer}')
for qubit in qubits.values():
circuit.append(cirq.X(qubit)**beta)
return circuit
def add_noise(circuit, noise_prob):
"""
Adds realistic quantum noise to the circuit.
Args:
circuit: A Cirq circuit
noise_prob: Probability of depolarizing noise
Returns:
A circuit with added noise operations
"""
if noise_prob <= 0:
return circuit
noisy_ops = []
for op in circuit.all_operations():
noisy_ops.append(op)
for q in op.qubits:
noisy_ops.append(cirq.DepolarizingChannel(noise_prob).on(q))
return cirq.Circuit(noisy_ops)
def evaluate_maxcut(bitstring, graph):
"""
Evaluates the MaxCut value for a given bitstring.
Args:
bitstring: String of '0's and '1's representing the cut
graph: NetworkX graph
Returns:
The value of the cut (number of edges between partitions)
"""
cut_value = 0
for i, j in graph.edges():
# Edge contributes to the cut if the endpoints are in different partitions
if bitstring[i] != bitstring[j]:
cut_value += 1
return cut_value
def run_qaoa(n_nodes=4, edge_probability=0.5, p_layers=1, noise_prob=0.0, num_samples=100):
"""
Runs the QAOA algorithm for the MaxCut problem.
Args:
n_nodes: Number of nodes in the graph
edge_probability: Probability of edge creation in random graph
p_layers: Number of QAOA layers (depth)
noise_prob: Probability of depolarizing noise
num_samples: Number of circuit samples to take
Returns:
Dictionary with QAOA results and visualization
"""
log = []
log.append("=== Quantum Approximate Optimization Algorithm (QAOA) Simulation ===")
# Create a random graph
graph = nx.gnp_random_graph(n_nodes, edge_probability, seed=42)
log.append(f"Created random graph with {n_nodes} nodes and {graph.number_of_edges()} edges")
log.append(f"Edges: {list(graph.edges())}")
# Create qubits
qubits = {i: cirq.NamedQubit(f'q{i}') for i in range(n_nodes)}
# Create QAOA circuit
circuit = create_qaoa_circuit(qubits, graph, p=p_layers)
log.append(f"Created QAOA circuit with p={p_layers} layers")
# Generate random circuit parameters (in a real QAOA, these would be optimized)
params = {}
for layer in range(p_layers):
# Typical ranges for parameters
params[f'gamma_{layer}'] = random.uniform(0, 2 * np.pi)
params[f'beta_{layer}'] = random.uniform(0, np.pi)
log.append("Generated random parameters for demonstration:")
for param, value in params.items():
log.append(f" {param} = {value:.4f}")
# Resolve the circuit with the parameters
resolved_circuit = cirq.resolve_parameters(circuit, params)
# Add noise if specified
if noise_prob > 0:
resolved_circuit = add_noise(resolved_circuit, noise_prob)
log.append(f"Added noise with probability {noise_prob}")
# Add measurements
measure_circuit = cirq.Circuit()
measure_circuit.append(cirq.measure(*qubits.values(), key='cut'))
# Combine circuits
full_circuit = resolved_circuit + measure_circuit
# Run the circuit
simulator = cirq.Simulator()
result = simulator.run(full_circuit, repetitions=num_samples)
# Process the measurement results
cut_counts = {}
cut_values = {}
for sample in result.measurements['cut']:
bitstring = ''.join([str(bit) for bit in sample])
if bitstring not in cut_counts:
cut_counts[bitstring] = 0
cut_values[bitstring] = evaluate_maxcut(bitstring, graph)
cut_counts[bitstring] += 1
# Find the best cut
best_bitstring = max(cut_values, key=cut_values.get)
best_cut_value = cut_values[best_bitstring]
best_count = cut_counts[best_bitstring]
log.append(f"\nSampling Results (from {num_samples} samples):")
for bitstring, count in sorted(cut_counts.items(), key=lambda x: cut_values[x[0]], reverse=True)[:5]:
log.append(f" Cut {bitstring}: value = {cut_values[bitstring]}, frequency = {count}/{num_samples} ({count/num_samples:.2%})")
log.append(f"\nBest cut found: {best_bitstring}")
log.append(f"Cut value: {best_cut_value}")
log.append(f"Frequency: {best_count}/{num_samples} ({best_count/num_samples:.2%})")
# Brute force the optimal solution for comparison
optimal_cut = 0
optimal_bitstring = None
for i in range(2**n_nodes):
bitstring = format(i, f'0{n_nodes}b')
cut_value = evaluate_maxcut(bitstring, graph)
if cut_value > optimal_cut:
optimal_cut = cut_value
optimal_bitstring = bitstring
log.append(f"\nOptimal cut (brute force): {optimal_bitstring}")
log.append(f"Optimal cut value: {optimal_cut}")
# Calculate approximation ratio
approx_ratio = best_cut_value / optimal_cut if optimal_cut > 0 else 1.0
log.append(f"Approximation ratio: {approx_ratio:.4f}")
# Generate circuit SVG for visualization
circuit_svg = circuit_to_svg(circuit)
# Generate simple text representation of the graph and cut
partition_visualization = "Graph partitioning based on best cut:\n"
partition_0 = []
partition_1 = []
for i in range(n_nodes):
if best_bitstring[i] == '0':
partition_0.append(i)
else:
partition_1.append(i)
partition_visualization += f" Partition 0: {partition_0}\n"
partition_visualization += f" Partition 1: {partition_1}\n\n"
partition_visualization += "Edge cuts:\n"
for i, j in graph.edges():
if best_bitstring[i] != best_bitstring[j]:
partition_visualization += f" ({i}, {j}) - Cut\n"
else:
partition_visualization += f" ({i}, {j}) - Not cut\n"
log.append(f"\n{partition_visualization}")
# Return results
return {
"n_nodes": n_nodes,
"n_edges": graph.number_of_edges(),
"p_layers": p_layers,
"noise_prob": noise_prob,
"num_samples": num_samples,
"best_cut": best_bitstring,
"best_cut_value": best_cut_value,
"best_frequency": best_count/num_samples,
"optimal_cut": optimal_bitstring,
"optimal_cut_value": optimal_cut,
"approximation_ratio": float(approx_ratio),
"params": params,
"circuit_svg": circuit_svg,
"graph_edges": list(graph.edges()),
"partition_0": partition_0,
"partition_1": partition_1,
"log": "\n".join(log)
}
if __name__ == '__main__':
# Run QAOA examples
qaoa_simple = run_qaoa(4, 0.5, p_layers=1)
qaoa_larger = run_qaoa(6, 0.4, p_layers=2)
qaoa_noisy = run_qaoa(5, 0.6, p_layers=1, noise_prob=0.01) |