Update app.py

app.py

@@ -110,7 +110,66 @@ result = simulator.run(compiled_circuit, shots=1024).result()
 counts = result.get_counts()
 print(counts)
 """,
-    "8. DNA Sequence Matching with Grover's Algorithm": """
+    "8. QUBO": """
+
+import numpy as np
+
+# Qiskit / Qiskit Optimization imports
+from qiskit import Aer
+from qiskit.utils import QuantumInstance
+from qiskit.algorithms import QAOA
+from qiskit.algorithms.optimizers import SPSA
+from qiskit_optimization import QuadraticProgram
+from qiskit_optimization.algorithms import MinimumEigenOptimizer
+
+# 1) Define a small QUBO with two binary variables
+problem = QuadraticProgram("my_qubo")
+problem.binary_var("x0")
+problem.binary_var("x1")
+
+# Objective: minimize H = x0 + 2*x1 + (1)*x0*x1
+#  - linear coefficients => x0: 1, x1: 2
+#  - quadratic coefficient => (x0, x1): 1
+problem.minimize(
+    linear={"x0": 1, "x1": 2},
+    quadratic={("x0", "x1"): 1}
+)
+
+# Print to confirm the problem is not empty
+print("\n--- Quadratic Program ---")
+print(problem.export_as_lp_string())
+
+# 2) Set up a QAOA solver on the qasm_simulator
+backend = Aer.get_backend('qasm_simulator')
+
+# Configure the quantum instance
+quantum_instance = QuantumInstance(
+    backend=backend,
+    shots=512,
+    seed_simulator=42,
+    seed_transpiler=42
+)
+
+# Create a QAOA instance
+qaoa = QAOA(
+    optimizer=SPSA(maxiter=50),  # or COBYLA(), NFT(), etc.
+    reps=2,                      # Number of QAOA layers
+    quantum_instance=quantum_instance
+)
+
+# Wrap QAOA in a MinimumEigenOptimizer
+solver = MinimumEigenOptimizer(qaoa)
+
+# 3) Solve the QUBO
+result = solver.solve(problem)
+
+# Print results
+print("\n--- QAOA Results ---")
+print("Optimal solution:", result.x)
+print("Objective value:", result.fval)
+
+    """,
+    "9. DNA Sequence Matching with Grover's Algorithm": """
 from qiskit import QuantumCircuit, transpile
 from qiskit_aer import AerSimulator
 from qiskit.circuit.library import GroverOperator