Add example: Qiskit BackendEstimatorV2 with MQSS adapter (#27)

* Add example: Qiskit BackendEstimatorV2 with MQSS adapter

* Apply pre-commit fixes

* Apply pre-commit fixes

* remove the transpilation

* remove the transpilation

Author: marhvera

examples/backend_estimator.py

@@ -0,0 +1,58 @@
+import math
+import sys
+from qiskit import QuantumCircuit
+from mqss.qiskit_adapter import MQSSQiskitAdapter
+from qiskit.quantum_info import SparsePauliOp, Statevector
+from qiskit.primitives import BackendEstimatorV2
+
+adapter = MQSSQiskitAdapter(token="<api-token>")
+[backend] = adapter.backends(name="<resource-name>")
+
+
+def estimate_required_shots(precision, variance=1.0):
+    if precision <= 0:
+        raise ValueError("precision must be > 0")
+    return math.ceil(variance / (precision**2))
+
+
+# State-preparation circuit
+qc = QuantumCircuit(2)
+qc.h(0)
+qc.cx(0, 1)
+
+# Define a simple Hamiltonian
+H = SparsePauliOp.from_list(
+    [
+        ("ZI", 1.0),
+        ("IZ", 1.0),
+        ("ZZ", 1.0),
+    ]
+)
+
+# Compute the ideal expectation for comparisons
+sv_logical = Statevector.from_instruction(qc)
+ev_ideal = sv_logical.expectation_value(H)
+ev_ideal_val = float(ev_ideal.real)
+
+# Run the Estimator grouping commuting observables
+estimator = BackendEstimatorV2(backend=backend)
+estimator.options.abelian_grouping = True
+precision = 0.01
+
+estimated_num_shots = estimate_required_shots(precision=precision, variance=1.0)
+print(f"Requested precision = {precision}")
+print(f"Estimated shots (placeholder worst-case) ≈ {estimated_num_shots}")
+
+if estimated_num_shots > 20000:
+    print("Error: estimated shot count is too high.")
+    print("Consider using a looser precision (e.g. precision=0.05).")
+    sys.exit(1)
+
+job = estimator.run([(qc, H)], precision=precision)
+result = job.result()
+pub_result = result[0]
+
+ev_backend_val = float(pub_result.data.evs)
+print("Ideal Estimator expectation value   =", ev_ideal_val)
+print("Backend Estimator expectation value =", ev_backend_val)
+print("Absolute error |backend - ideal|   =", abs(ev_backend_val - ev_ideal_val))