This quantum circuit demonstrates the three-qubit bit-flip error correction code.
Here is a new quantum circuit that implements the three-qubit bit-flip error correction code, which is not part of the standard algorithms you've listed. This circuit encodes a qubit into three qubits to protect against bit-flip errors, introduces an error on one qubit, detects the error using syndrome measurements, and shows that the error can be identified and corrected.
**Description:**
This quantum circuit demonstrates the three-qubit bit-flip error correction code. The circuit starts by preparing a qubit in the superposition state \(|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)\). It then encodes this logical qubit into three physical qubits using two CNOT gates to protect against bit-flip errors. A bit-flip error (X gate) is intentionally introduced on the second qubit. The circuit performs syndrome measurements by applying additional CNOT gates and measures two of the qubits to detect the presence and location of the error. Finally, it measures the logical qubit, demonstrating that the error can be detected and, in practice, corrected based on the measurement outcomes.
```python
from qiskit import QuantumCircuit
# Create a circuit with 3 qubits and 3 classical bits
qc = QuantumCircuit(3, 3)
# Prepare the logical qubit |+⟩ = (|0⟩ + |1⟩)/√2 on qubit 0
qc.h(0)
# Encode qubit 0 into qubits 0, 1, and 2 using the bit-flip code
qc.cx(0, 1)
qc.cx(0, 2)
# Introduce a bit-flip error on qubit 1
qc.x(1)
# Perform syndrome measurements to detect the error
qc.cx(0, 1)
qc.cx(0, 2)
# Measure qubits 1 and 2 to obtain the error syndrome
qc.measure(1, 1)
qc.measure(2, 2)
# Measure the logical qubit
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"}]}\''