Algorithm guide · protocol
Entanglement swapping
Entangle two qubits that never interacted, by measuring the two in the middle. The repeater primitive.
Circuit size: 4 qubits, 4 classical bits
Two Bell pairs · Bell measurement · Conditional versus unconditional states
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
Entangle distant endpoint qubits q0 and q3 by Bell-measuring their partners q1 and q2. The endpoints never interact directly. Identify the conditional Bell state and optionally correct it to Φ+.
2. How it works
The original pairings are not the only way to describe the four-qubit state. A middle Bell measurement selects a corresponding Bell state for the endpoints. Its classical record specifies which Pauli adjustment is needed.
Prepare
Create Bell pairs (q0,q1) and (q2,q3).
Middle Bell measurement
Apply CX q1→q2 and H q1; record q1→c0 and q2→c1.
Correct endpoints
Apply the selected classical X/Z feed-forward to align the remote pair with Φ+.
Verify in a basis
Measure endpoint q0→c2 and q3→c3 in Z or X.
3. The mathematics, one idea at a time
The sum runs over four Bell labels. Conventions can change phases and byproduct order, but every middle label has probability 1/4.
Endpoints agree in both the Z and X basis experiments. Before correction, ignoring the middle record leaves their joint state I/4.
Work this through by hand
4. Corrections on, Z readout
Four fixed qubits: two endpoints and two middle partners.
Choose Corrections on · Z readout.
| Stage | Calculation or state | What it means |
|---|---|---|
| Middle outcome | Each c1c0 has probability 1/4 | No particular Bell branch is postselected. |
| After correction | Endpoint 00 or 11 each has probability 1/2 | This holds in every middle branch. |
| Joint record | Endpoint bits c3c2 precede middle bits c1c0 | There are eight allowed full strings, each probability 1/8. |
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 |
|---|---|---|
| 00mm and 11mm, for mm=00,01,10,11 | 12.5% per full string ideally | Endpoints agree while the middle record varies. |
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: Entanglement swapping · corrections on · Z readout
- 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
Split c3c2c1c0 as endpoint pair | middle record. Sum over the middle bits to get the remote endpoint histogram.
With correction off, endpoint Z parity equals c1 and endpoint X parity equals c0 in their respective experiments.
Endpoint measurement consumes the pair. To compare Z and X, run fresh preparations; do not measure both bases sequentially on the same pair.
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
- Ignoring the middle result without correcting is not the same as having a known pure Bell state.
- A single agreeing Z histogram alone does not certify entanglement.
8. Check your understanding
Without correction, what Z endpoint parity is expected if c1=1?
Reveal the reasoning
Odd parity: endpoint outcomes differ. The middle record tells which Bell correlation branch occurred.
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.
- Entanglement swapping · corrections on · Z readout
- Entanglement swapping · corrections on · X readout
- Entanglement swapping · corrections off · Z readout
- Entanglement swapping · corrections off · X readout
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.