Lab 2 — Quantum Teleportation

Overview

Quantum teleportation moves the state of a qubit from one place (Alice) to another (Bob) without physically sending the qubit and without ever learning what the state is. It does not violate the no-cloning theorem — Alice's original qubit is destroyed in the process — and it does not transmit information faster than light, because Bob's qubit is useless until he receives two classical bits from Alice.

This is a cornerstone protocol: it underlies quantum repeaters, distributed quantum computing, and many error-correction schemes. In this lab you prepare an arbitrary state, teleport it, and confirm Bob ends up holding exactly what Alice started with.

Theory

Alice wants to send the unknown state

$$\lvert \psi \rangle = \alpha \lvert 0 \rangle + \beta \lvert 1 \rangle .$$

Alice and Bob first share an entangled Bell pair, created by a Hadamard followed by a CNOT:

$$\lvert \Phi^+ \rangle = \frac{1}{\sqrt{2}}\left( \lvert 00 \rangle + \lvert 11 \rangle \right).$$

The full three-qubit system is $\lvert \psi \rangle \otimes \lvert \Phi^+ \rangle$. Alice applies a CNOT from her message qubit onto her half of the Bell pair, then a Hadamard on the message qubit, then measures both of her qubits. Expanding the algebra, the global state can be rewritten as

$$\frac{1}{2}\Big[ \lvert 00 \rangle (\alpha\lvert 0\rangle + \beta\lvert 1\rangle) + \lvert 01 \rangle (\alpha\lvert 1\rangle + \beta\lvert 0\rangle) + \lvert 10 \rangle (\alpha\lvert 0\rangle - \beta\lvert 1\rangle) + \lvert 11 \rangle (\alpha\lvert 1\rangle - \beta\lvert 0\rangle) \Big].$$

The two classical bits $(m_1, m_0)$ Alice measures tell Bob which of the four cases occurred. Bob then applies a correction:

Alice's bits $(m_1 m_0)$	Bob's qubit	Correction
00	$\alpha\lvert 0\rangle + \beta\lvert 1\rangle$	$I$ (nothing)
01	$\alpha\lvert 1\rangle + \beta\lvert 0\rangle$	$X$
10	$\alpha\lvert 0\rangle - \beta\lvert 1\rangle$	$Z$
11	$\alpha\lvert 1\rangle - \beta\lvert 0\rangle$	$ZX$

After the correction Bob's qubit is exactly $\lvert \psi \rangle$.

Implementation

We use three qubits: q0 is Alice's message, q1 is Alice's half of the Bell pair, q2 is Bob's half. We teleport a specific test state and then invert its preparation on Bob's qubit; if teleportation worked, Bob's qubit returns to $\lvert 0 \rangle$ and we should measure 0 every time.

import numpy as np
from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, transpile
from qiskit_aer import AerSimulator

def build_teleportation(theta: float, phi: float) -> QuantumCircuit:
    """Teleport Ry(theta)Rz(phi)|0> from q0 to q2, then verify."""
    q = QuantumRegister(3, "q")      # q0 message, q1 Alice's pair, q2 Bob's pair
    c = ClassicalRegister(3, "c")    # c0,c1 Alice's measurements; c2 final check
    qc = QuantumCircuit(q, c)

    # 1. Prepare an arbitrary state |psi> on Alice's qubit q0.
    qc.ry(theta, q[0])
    qc.rz(phi, q[0])
    qc.barrier()

    # 2. Create the Bell pair across q1 (Alice) and q2 (Bob).
    qc.h(q[1])
    qc.cx(q[1], q[2])
    qc.barrier()

    # 3. Alice entangles her message with her half, then rotates to the X basis.
    qc.cx(q[0], q[1])
    qc.h(q[0])
    qc.barrier()

    # 4. Alice measures her two qubits.
    qc.measure(q[0], c[0])  # m0 -> controls X correction
    qc.measure(q[1], c[1])  # m1 -> controls Z correction

    # 5. Bob applies classically-conditioned corrections.
    with qc.if_test((c[1], 1)):
        qc.z(q[2])
    with qc.if_test((c[0], 1)):
        qc.x(q[2])
    qc.barrier()

    # 6. Verification: undo the original preparation on Bob's qubit.
    #    If teleportation is perfect, q2 returns to |0>.
    qc.rz(-phi, q[2])
    qc.ry(-theta, q[2])
    qc.measure(q[2], c[2])
    return qc

if __name__ == "__main__":
    theta, phi = 1.2345, 0.7  # an arbitrary, generic state
    qc = build_teleportation(theta, phi)

    sim = AerSimulator()
    result = sim.run(transpile(qc, sim), shots=4096).result()
    counts = result.get_counts()

    # Aggregate over Alice's (random) bits, looking only at c2 (leftmost char).
    bob = {"0": 0, "1": 0}
    for bitstring, n in counts.items():
        bob[bitstring[0]] += n
    print("Bob's verification qubit:", bob)
    success = bob["0"] / sum(bob.values())
    print(f"Teleportation fidelity (ideal=1.0): {success:.4f}")

Key points:

• with qc.if_test((c[i], 1)): is the modern Qiskit way to apply a gate conditioned on a classical measurement result — exactly the classical channel from Alice to Bob.
• The barrier() calls are visual separators that prevent the transpiler from merging the protocol stages; they do not change the result.
• The verification trick (steps 6) is the cleanest way to check teleportation without statevector access: we apply the inverse of the preparation and expect 0.

Run it

Because Alice's measured bits are genuinely random, the leftmost character of each bitstring (Bob's check) should still be 0 in every shot:

Bob's verification qubit: {'0': 4096, '1': 0}
Teleportation fidelity (ideal=1.0): 1.0000

A fidelity of 1.0000 on the noiseless simulator confirms the state arrived intact. On real hardware you would see a value slightly below 1 due to gate and measurement noise.

Exercises

1. (Beginner) Print the full counts dictionary and confirm that Alice's two bits (c0, c1) appear with roughly equal 25% frequency across the four combinations.
2. (Beginner) Teleport the basis state $\lvert 1 \rangle$ (set theta = np.pi, phi = 0) and remove the verification step. What does Bob's qubit measure as, and why?
3. (Intermediate) Use qiskit.quantum_info.Statevector to confirm the Bell pair $\lvert \Phi^+ \rangle$ is created correctly before any teleportation steps.
4. (Intermediate) Replace the two if_test corrections with the deferred-measurement trick: use cz and cx gates controlled on the qubits directly (no measurement) and explain why the result is equivalent.
5. (Advanced) Add a depolarizing noise model to the AerSimulator and plot the teleportation fidelity as a function of the error rate.

Further reading

• Bennett et al., Teleporting an Unknown Quantum State via Dual Classical and EPR Channels, Phys. Rev. Lett. 70, 1895 (1993) — the original paper.
• Qiskit textbook: Quantum Teleportation.
• Previous lab: Quantum Random Number Generator. Next: Grover Search.