Algorithm guide · protocol
Quantum Teleportation
Transmit a qubit's state using a shared Bell pair and two classical bits.
Circuit size: 3 qubits, 2 classical bits
Bell pairs · Bell measurement · Classically controlled X/Z
Review quantum fundamentals →Notation used in this lesson
- |ψ⟩ and ⟨ψ|
- A state vector (ket) and its conjugate transpose (bra). ⟨a|b⟩ is their inner product.
- α, β and i
- α and β are complex amplitudes; i²=−1. A probability is the squared magnitude |α|².
- XOR, · and ⊗
- XOR adds bits modulo two. In s·x, multiply corresponding bits and XOR them. ⊗ combines registers or operations on different wires.
- U†, Σ and Tr
- U† is the adjoint (inverse for a unitary). Σ sums terms. Tr is the matrix trace, the sum of diagonal entries.
- Probability, shots and angles
- Ideal probabilities describe the model; measured shot frequencies fluctuate. Angles inside trigonometric expressions are in radians unless explicitly labeled turns.
1. The problem
Transfer an unknown qubit state from Alice’s input to Bob’s output using shared entanglement and two classical bits. The input is consumed; teleportation does not create a second copy.
2. How it works
A Bell measurement distributes the input amplitudes into four possible Pauli-related versions on Bob. The random two-bit record tells Bob which reversible correction restores the original state.
Prepare resource
q0 is the input; create a Bell pair on q1,q2. Bob owns q2.
Bell-analyze
Apply CX q0→q1, then H on q0.
Measure and send
Measure q0→c0 and q1→c1. Each two-bit result is equally likely.
Correct Bob
Apply X if c1=1 and Z if c0=1. q2 then holds the input state.
3. The mathematics, one idea at a time
The protocol must preserve arbitrary complex amplitudes, not just one classical bit.
Corrections cancel the random Pauli byproducts. Their order changes only a global phase for this purpose.
Work this through by hand
4. Warm-up |0⟩; worked coherent input |+⟩
Three fixed qubits and two classical bits. |0⟩ is the simplest basis check; |+⟩ is the smallest equally sized example that tests coherence.
Select Teleport |+⟩.
| Stage | Calculation or state | What it means |
|---|---|---|
| Input | α=β=1/√2 | Bob must receive a plus relative sign, not just a 50/50 Z histogram. |
| Bell branches | c1c0=00,01,10,11 each with probability 1/4 | The classical record is independent of the unknown amplitudes. |
| After feed-forward | q2=|+⟩ in every branch | Add H then measurement on q2 to verify a certain 0. |
Expected outcomes for this example
These are ideal predictions for the stated configuration, not a result already measured by your browser. Rows describing alternative modes or combined events are labeled explicitly.
| Outcome or event | Ideal probability | Interpretation |
|---|---|---|
| 00,01,10,11 | 25% each ideally | Random Bell record; not the teleported state. |
5. Load the worked example in Composer
The worked example can use smaller or simpler inputs than the gallery defaults. The button below loads the exact circuit used in this lesson.
Selected variant: Teleport |+>
- Save any current circuit you want to keep, then open the worked example. Loading it does not execute a paid hardware job.
- Trace the preparation, main operations and measurements against the worked table. Check the gate editor for controls, targets and angles.
- Start with an ideal noiseless simulator compatible with the circuit. Mid-circuit measurement, reset and feed-forward require support from the chosen backend.
- Inspect the live preview. For a recorded experiment, open Execute, set the available backend options and shots, and Run. Sign in when prompted to retain it in My Jobs.
- Read the output as described below before changing one setting and repeating. The guide describes the original example; edits can change the answer.
6. Read the output correctly
Classical bits records the two consumed Alice wires. Bob is intentionally unmeasured in the template.
Inspect Bob’s conditional state, or verify in the appropriate basis. A direct Z histogram of |+⟩ is flat and does not distinguish |+⟩ from |−⟩.
Use a backend supporting mid-circuit measurement and classical feed-forward.
A bit label is a register value; its bar height is a probability or count. For S shots and observed count k, the sampled frequency is k/S. A typical sampling scale for probability p is √[p(1−p)/S]; this is not a strict error bound. Finite samples do not have to match the ideal percentages exactly.
7. Common mistakes & limits
- Two transmitted classical bits do not contain a classical description of α and β.
- Without classical communication Bob’s local state does not reveal Alice’s input. Teleportation does not signal faster than light.
8. Check your understanding
For input |+⟩ and a=1,b=0, what state reaches Bob before correction?
Reveal the reasoning
Z|+⟩=|−⟩. Applying the recorded Z correction restores |+⟩.
9. Further study
The lesson’s register mappings, supported inputs and loadable example follow Quantket’s implemented circuit. A reference may use different wire names, bit order or resource conventions.
Gallery reference: defaults, definitions & implementation notes
Available example inputs
Choose a named circuit variant in Algorithms. Each variant fixes its preparation, operations and measurement mapping; the worked variant above is one of them.
- Teleport |0>
- Teleport |1>
- Teleport |+>
- Teleport |->
- Teleport |+i>
- Teleport custom
Read your saved result
Check the submitted circuit, measured register and execution settings in Run details. Exact results give computed probabilities; sampled results show frequencies from the recorded shots. Edited circuits and noise can change the expected outcome. The algorithm label records the circuit's origin; it does not certify an edited circuit or a measured result.