A three-qubit circuit can transfer an unknown quantum state from Alice’s qubit to Bob’s qubit in a simulator. The recipe is: prepare a Bell pair, entangle the input with Alice’s half, measure Alice’s two qubits, send those two classical bits, and conditionally apply Pauli corrections to Bob’s qubit. To test the result with a known input, undo the input preparation on Bob’s qubit and measure; an ideal simulator returns 0 on that verification bit.
This is state transfer, not movement of matter. The protocol consumes shared entanglement, requires ordinary classical communication, does not clone the input, and cannot send information faster than light.
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What quantum teleportation actually transfers
Quantum teleportation moves the state carried by one qubit onto a distant qubit without measuring the unknown state directly. The original state is destroyed by Alice’s measurement, so there are not two copies at the end. Bob receives the state only after he gets Alice’s two classical measurement results and performs the matching correction.
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The standard demonstration needs three qubits:
| Qubit | Role | What happens to it |
|---|---|---|
| 0 | Input (Alice’s secret state) | Prepared in the state to be teleported, then measured by Alice |
| 1 | Alice’s half of an entangled pair | Entangled with qubit 2, then measured with qubit 0 |
| 2 | Bob’s half of the entangled pair | Receives a conditional X and/or Z correction |
Qubits 1 and 2 form one shared e-bit. Creating that pair is the resource that makes teleportation possible; it is consumed during the Bell measurement. Alice’s two classical bits are not optional metadata: without them Bob cannot know which correction is required.
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What you need for a local simulator
- Python 3 and a virtual environment are recommended.
- Qiskit for circuit construction.
- A simulator backend such as Qiskit Aer. A local simulator is sufficient; no quantum-computing account or physical processor is required for this walkthrough.
- A known test state. Using a known preparation lets you verify the transfer deterministically instead of trying to compare an unknown state directly.
Install the packages in the environment you intend to use:
python -m pip install qiskit qiskit-aerQiskit syntax for conditional operations has changed across releases. The circuit logic below is stable, but use the conditional form supported by your installed version; newer releases provide an if_test context, while older releases commonly use c_if.
The circuit in conceptual order
1. Prepare the input
Choose a state that is easy to prepare and easy to undo. For example, applying Ry(theta) to qubit 0 creates a test state that depends on the angle theta. The protocol itself does not require the state to be known; the known preparation is only for verification.
2. Create the Bell pair
Apply a Hadamard gate to qubit 1, then a controlled-NOT from qubit 1 to qubit 2. In symbols, this creates the entangled state usually written as (|00> + |11>) / sqrt(2) for those two qubits.
qc.h(1)
qc.cx(1, 2)3. Perform Alice’s Bell-measurement transformation
Entangle the input with Alice’s Bell-pair half using a CNOT from qubit 0 to qubit 1, then apply a Hadamard to qubit 0:
qc.cx(0, 1)
qc.h(0)After these gates, measuring qubits 0 and 1 in the computational basis produces two classical bits. The measurement outcomes select which Pauli operation Bob must apply.
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4. Measure Alice’s qubits
For the register convention used here, qubit 0 is measured into classical bit 0 and qubit 1 into classical bit 1:
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qc.measure(1, 1)5. Apply Bob’s correction
With that exact wiring, apply Z to Bob’s qubit when classical bit 0 is 1, and apply X when classical bit 1 is 1. Both conditions can be true, in which case Bob applies both corrections.
| Measured bit 0 (from q0) | Measured bit 1 (from q1) | Operation on q2 |
|---|---|---|
| 0 | 0 | Identity |
| 0 | 1 | X |
| 1 | 0 | Z |
| 1 | 1 | X and Z (order changes only a global phase) |
The labels “bit 0” and “bit 1” describe this circuit’s classical destinations, not an immutable rule about every program. Other examples place the input in a different qubit, reverse control and target wires, or map measurement results into a register whose displayed order is reversed. Always derive the correction from your own measurement mapping.
A complete Qiskit example with inverse-preparation verification
The following example uses Ry(theta) as the known input preparation. It measures Alice’s qubits, performs real-time feed-forward in the simulator, undoes the same rotation on Bob’s qubit, and measures a verification bit. In an ideal run, that final bit is always 0.
from qiskit import QuantumCircuit
from qiskit_aer import AerSimulator
# Any angle is suitable for this known test state.
theta = 0.73
qc = QuantumCircuit(3, 3)
# 1. Prepare |psi> = Ry(theta)|0> on the input qubit.
qc.ry(theta, 0)
# 2. Make the Bell pair shared by Alice (q1) and Bob (q2).
qc.h(1)
qc.cx(1, 2)
# 3. Alice's Bell-measurement transformation.
qc.cx(0, 1)
qc.h(0)
# 4. Alice measures q0 -> c0 and q1 -> c1.
qc.measure(0, 0)
qc.measure(1, 1)
# 5. Conditional corrections for this exact mapping.
# Newer Qiskit releases:
with qc.if_test((qc.clbits[0], 1)):
qc.z(2)
with qc.if_test((qc.clbits[1], 1)):
qc.x(2)
# 6. Verification only: undo the known preparation on Bob.
qc.ry(-theta, 2)
qc.measure(2, 2)
simulator = AerSimulator()
result = simulator.run(qc, shots=1024).result()
print(result.get_counts())If your installed Qiskit version does not support if_test, express the same two conditions with that release’s classical-control API, commonly qc.z(2).c_if(qc.clbits[0], 1) and qc.x(2).c_if(qc.clbits[1], 1). The important part is not the spelling of the method; it is that the condition tests the classical bit that received each measurement.
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Why the inverse gate proves the transfer
After the corrections, Bob should hold exactly the state that was prepared on qubit 0. Directly comparing two arbitrary quantum states is not possible with one measurement, so the test uses a reversible “uncompute” operation.
- Prepare the known state with a gate sequence
Uon qubit 0. - Run teleportation and corrections.
- Apply
U-dagger, the inverse of that preparation, to Bob’s qubit. - Measure Bob. A correctly transferred state becomes
|0>, so the verification bit should be 0.
For the example, U is Ry(theta) and U-dagger is Ry(-theta). The inverse rotation is not part of teleportation. It is a diagnostic added after the protocol; omitting it is correct when you want Bob’s state itself rather than a pass/fail test.
Statevector simulation versus shot simulation
Statevector view
A statevector simulator represents the complete ideal amplitudes. It is useful for inspecting how the three-qubit state evolves before measurement and for checking that Bob’s reduced state matches the prepared input after the correction. Statevectors expose amplitudes and relative phase directly, but measurements and classical feed-forward require care: measuring Alice’s qubits collapses the state, so inspect the appropriate point in the circuit or use a simulator feature that saves the statevector before the final measurements.
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Shot-based view
A shot-based run executes the circuit repeatedly and returns a histogram of classical bit strings. This is closer to how an experiment reports data and works naturally with mid-circuit measurements and conditional operations. For the inverse-preparation test, every ideal shot has verification bit 0, although the two Bell-measurement bits are randomly distributed.
Do not interpret the random distribution of Alice’s two bits as a failure. Those bits carry the correction information. The pass/fail signal is the verification bit measured after Bob’s correction and inverse preparation.
Understanding the correction rule mathematically
Let the input be |psi>, and let Alice’s two measurement results be m0 and m1 under the wiring used above. Before correction, Bob’s qubit is one of four related states. The correction is represented compactly as X^m1 Z^m0 (up to an irrelevant global phase). Thus:
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m0 = 0, m1 = 0: no correction.m0 = 0, m1 = 1: applyX.m0 = 1, m1 = 0: applyZ.m0 = 1, m1 = 1: apply bothXandZ.
This relation explains why a swapped classical-register convention can make apparently reversed code still look plausible while producing the wrong state. The correction exponents must follow the actual destinations of the measured bits.
Common register-order mistakes
Reading a count string backwards
Many Qiskit count displays write the highest classical-bit index first. If you measure q0 to c0 and q1 to c1, a displayed string such as 010 should be read as c2=0, c1=1, c0=0. It is not evidence that the measurements occurred in the order shown on screen.
Copying a conditional line from another circuit
An instructional example may assign the input to q2, use a different classical register, or measure into reversed destinations. Copying its X/Z conditions without copying its wiring changes the protocol. Write a small mapping table beside your circuit before adding conditions.
Applying corrections before measurement
Bob’s correction depends on classical data that does not exist until Alice measures. Applying a fixed X or Z before the measurement is a different circuit and will only work for selected inputs by coincidence.
Troubleshooting a failed verification
| Symptom | Likely cause | Fix |
|---|---|---|
| Verification bit is sometimes 1 in an ideal simulator | A correction condition is attached to the wrong classical bit, or an X/Z assignment is swapped. |
Confirm q0->c0 and q1->c1 (or your chosen mapping), then rebuild the four-row correction table. |
| All counts look different from the example | Classical bit strings are displayed in reverse index order, or the input angle is different. | Decode the string by classical-bit index and check the leading verification bit separately. |
The circuit raises an error at if_test |
Your Qiskit version uses an older control-flow API. | Use the version-appropriate c_if form, keeping the same bit-to-condition mapping. |
| Bob’s qubit is measured before the correction | The final measurement was placed too early, or the simulator was given a circuit without dynamic feed-forward. | Keep Alice’s measurements, conditional operations, inverse preparation, and Bob’s verification measurement in that order. |
| A statevector comparison appears to differ by a sign | Two states may differ only by a global phase, which has no measurement consequence. | Use probabilities, a density-matrix comparison, or inverse-preparation measurement rather than raw amplitude equality alone. |
| Verification fails only after adding noise | Noise, readout errors, and imperfect gates alter real or noisy-simulator outcomes. | Separate protocol logic from hardware performance; first confirm an ideal run, then characterize noise with repeated shots. |
What the simulator demonstrates—and what it does not
An ideal simulator executes the mathematical circuit exactly, so a perfect verification result confirms that the gate sequence and classical mapping implement teleportation for the chosen input. It does not demonstrate that a physical processor will produce the same histogram. Real devices add gate error, decoherence, readout error, and timing constraints.
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Why teleportation does not violate physics
No faster-than-light message
Bob’s qubit has a state correlated with Alice’s measurement, but Bob cannot select Alice’s random result or decode the input before receiving the two classical bits. The classical channel is limited by ordinary communication speed, so teleportation cannot be used for instantaneous messaging.
No cloning
A measurement on Alice’s two qubits consumes the original input relationship. The protocol transfers the state rather than leaving an intact copy on Alice’s side. This is consistent with the no-cloning principle.
Entanglement is a resource, not a signal
The pre-shared Bell pair establishes correlations, but entanglement alone does not carry a controllable message. The classical correction completes the transfer and consumes the shared pair in the process.
Extending the experiment safely
Try other single-qubit preparations
Replace Ry(theta) with a sequence such as Rz(phi) followed by Ry(theta), then apply the exact inverse sequence in reverse order on Bob. For gates A followed by B, the inverse is B-dagger followed by A-dagger. Testing several angles and phases helps reveal a correction that happens to work for one real-valued state but fails for states with complex phase.
Inspect each correction branch
For debugging, run circuits that force or post-select each pair of Alice outcomes and verify that the corresponding Bob correction restores the input. This isolates the four branches of the protocol instead of hiding a single bad branch inside an aggregate histogram.
Separate protocol and verification circuits
Keep a copy that ends immediately after Bob’s correction when you want to study the teleported state, and a second copy that adds inverse preparation for a binary test. Mixing the diagnostic gate into a circuit intended to represent the protocol can make the final state difficult to interpret.
Minimal checklist before you run
- There are exactly three logical qubits: input, Alice, and Bob.
- The Bell pair uses
Hon Alice’s pair qubit followed byCXto Bob. - The Bell-measurement transformation is
CX(input, Alice)followed byH(input). - Both Alice qubits are measured before Bob’s conditional operations.
- Your correction table matches the actual classical-bit destinations.
- The inverse of the known input preparation is applied only for verification.
- The verification measurement is placed after correction and should be 0 for an ideal simulator.
- Any nonzero verification result is investigated as a mapping, ordering, API, or noise issue before drawing conclusions.
Once this three-qubit circuit works, you have the core pattern used in larger protocols: prepare entanglement, perform a joint measurement, communicate classical outcomes, and apply feed-forward corrections. The simulator makes every step inspectable while preserving the essential distinction between quantum state transfer and classical communication.
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