This circuit estimates the expectation value of a Pauli-Z operator using the Hadamard test.
```python
from qiskit import QuantumCircuit
# This circuit implements the Hadamard test to estimate the expectation value of a unitary operator U.
# Prepare a quantum circuit with 2 qubits (1 ancilla qubit and 1 target qubit) and 1 classical bit for measurement
qc = QuantumCircuit(2, 1)
# Step 1: Prepare the target qubit in a superposition state |ψ⟩
qc.h(1) # Apply Hadamard gate to qubit 1 to create |ψ⟩ = (|0⟩ + |1⟩)/√2
# Step 2: Prepare the ancilla qubit
# The ancilla qubit (qubit 0) is already initialized to |0⟩
# Step 3: Apply Hadamard gate to the ancilla qubit
qc.h(0)
# Step 4: Apply the controlled-U operation, where U is a Pauli-Z gate in this example
qc.cz(0, 1) # Controlled-Z gate with ancilla as control and target qubit as target
# Step 5: Apply another Hadamard gate to the ancilla qubit
qc.h(0)
# Step 6: Measure the ancilla qubit to obtain the real part of ⟨ψ|U|ψ⟩
qc.measure(0, 0)
```'Failed to run program: \'409 Client Error: Conflict for url: https://api.quantum.ibm.com/runtime/jobs. {"errors":[{"message":"You have reached the limit of 3 pending jobs. Please wait for a job to complete or cancel one before submitting anything new.","code":3458,"solution":"Wait until some previous jobs were finished. You can cancel pending jobs to run new jobs.","more_info":"https://docs.quantum-computing.ibm.com/errors"}]}\''