This circuit prepares an arbitrary state on Bob's qubit using entanglement and a single classical bit of information sent by Alice.
```python
from qiskit import QuantumCircuit
# Create a quantum circuit with 2 qubits and 1 classical bit
qc = QuantumCircuit(2, 1)
# Prepare an entangled Bell state between qubit 0 (Alice) and qubit 1 (Bob)
qc.h(0)
qc.cx(0, 1)
# Alice wants to remotely prepare a qubit in state |ψ⟩ = cos(θ)|0⟩ + sin(θ)|1⟩ on Bob's side
# Let's set θ = π/4 for this example
theta = 3.141592653589793 / 4
# Alice encodes the desired state onto her qubit
qc.ry(2 * theta, 0)
# Alice measures her qubit in the computational basis
qc.measure(0, 0)
# Bob applies a correction based on Alice's measurement result
qc.z(1).c_if(qc.clbits[0], 1)
```
This quantum circuit implements **Remote State Preparation (RSP)**. In this protocol, Alice knows the exact quantum state she wants to prepare on Bob's qubit. They share an entangled pair, and by performing a rotation and measurement on her qubit, Alice can remotely prepare the desired state on Bob's qubit by sending him one classical bit of information.
**Explanation:**
- **Entanglement Preparation:**
- Qubits 0 and 1 are entangled using a Hadamard gate followed by a CNOT gate.
- This creates a Bell state: \(|\Phi^+\rangle = \frac{1}{\sqrt{2}} (|00\rangle + |11\rangle)\).
- **State Encoding by Alice:**
- Alice applies an `RY` rotation to her qubit to encode the desired state. For θ = π/4, the rotation transforms her qubit.
- **Measurement by Alice:**
- Alice measures her qubit. The result is stored in the classical bit `qc.clbits[0]`.
- **Conditional Operation by Bob:**
- If Alice's measurement result is `1`, Bob applies a Pauli-Z gate to his qubit.
- This conditional operation ensures that Bob's qubit is in the desired state \(|ψ⟩\).
After this protocol, Bob's qubit is in the state \(|ψ⟩ = \cos(θ)|0⟩ + \sin(θ)|1⟩\) without any quantum communication of the state itself, demonstrating the power of entanglement in quantum information transfer.'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"}]}\''