This quantum circuit implements the HHL algorithm to solve a system of linear equations.
```python
from qiskit import QuantumCircuit
import numpy as np
# Create a quantum circuit with 4 qubits and 1 classical bit
# Qubits 0 and 1 for quantum phase estimation, qubit 2 for the system qubit, qubit 3 for the ancilla
qc = QuantumCircuit(4, 1)
# Initialize the system qubit |2⟩ to |1⟩ (representing the vector |b⟩ in Ax = b)
qc.initialize([0, 1], 2)
# Apply Hadamard gates to qubits 0 and 1 for superposition in the phase estimation register
qc.h([0, 1])
# Controlled unitary operations (simulate e^{iAt})
# For simplicity, use powers of a Pauli-Z gate to represent eigenvalues
qc.cp(np.pi/2, 0, 2) # Controlled phase rotation with eigenvalue 1 (π/2)
qc.cp(np.pi, 1, 2) # Controlled phase rotation with eigenvalue 2 (π)
# Apply inverse Quantum Fourier Transform to qubits 0 and 1
qc.h(1)
qc.cp(-np.pi/2, 0, 1)
qc.h(0)
# Perform rotation on the ancilla qubit conditioned on qubits 0 and 1
# Simulating the division by the eigenvalues
qc.crx(2 * np.arcsin(1/2), 0, 3)
qc.crx(2 * np.arcsin(1/4), 1, 3)
# Uncompute the phase estimation to disentangle qubits 0 and 1 from the system qubit
qc.h(0)
qc.cp(np.pi/2, 0, 1)
qc.h(1)
# Measure the ancilla qubit to obtain the solution probability
qc.measure(3, 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"}]}\''