Algorithm guide · protocol
Remote state preparation
Prepare a known polar or equatorial state on Bob's qubit using one shared Bell pair and one classical bit.
Circuit size: 2 qubits; 1 classical bit
Bell state · Known versus unknown input states · Measurement feed-forward
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
Alice knows a target state from an agreed equatorial or real polar great circle. Prepare that state on Bob’s qubit using one shared Bell pair and one classical bit. The one-bit guarantee is restricted to these ensembles.
2. How it works
Alice chooses a target-dependent measurement basis. Her result steers Bob into either the target or a related orthogonal state. Because the ensemble is restricted, one fixed correction for each circle converts the second branch into the target.
Share Φ+
Prepare (|00⟩+|11⟩)/√2 on Alice q0 and Bob q1.
Alice measures
Apply the target-dependent local rotation and measurement, recording c0.
Communicate one bit
Bob learns whether the correcting operation is needed.
Correct and inspect
If c0=1, apply Z for the equatorial circle or Y for the polar circle. Leave Bob unmeasured.
3. The mathematics, one idea at a time
The state’s probabilities in Z remain 1/2; its angle changes relative phase.
This real-amplitude great circle admits a different fixed orthogonal-branch correction.
Bob cannot read the target from his uncorrected local state without classical communication.
Work this through by hand
4. Prepare |+⟩ on Bob
Two qubits and one classical bit, the fixed minimum resource for this implementation.
Select Equatorial, angle 0 turns and correction on.
| Stage | Calculation or state | What it means |
|---|---|---|
| Target | φ=0 ⇒ |ψ⟩=|+⟩ | This avoids complex phase for the first example. |
| Alice’s result | c0=0 gives Bob |+⟩; c0=1 gives |−⟩ | Both records occur with probability 1/2. |
| Bob’s correction | Z|−⟩=|+⟩ | Both branches now have the same target, up to global phase. |
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 |
|---|---|---|
| 0 | 50% ideal | Alice’s record; no correction needed. |
| 1 | 50% ideal | Alice’s record; Z correction is applied. |
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.
- Target circle
- Equatorial
- Angle (turns)
- 0
- Apply Bob's correction
- On
- 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 shows Alice’s random record, not whether Bob succeeded. Inspect q1’s state or its Bloch representation after correction.
For this |+⟩ example, append H to Bob and measure him to obtain 0 ideally. Direct Z measurement would give 0/1 equally and would not establish the + phase.
Try 0.25 turns next: the equatorial target becomes |+i⟩.
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
- Alice knows the target here; this is not teleportation of an arbitrary unknown input.
- One classical bit does not suffice for this exact deterministic protocol over all possible qubit states.
8. Check your understanding
If correction is off and you ignore c0, what is Bob’s state?
Reveal the reasoning
I/2, an equal mixture of |+⟩ and |−⟩ in this example. It is not a pure |+⟩ state.
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
Inputs & default settings
These are the gallery defaults. The worked example above may override them. Configure the card to change the inputs before loading its editable circuit.
- Target circle
- Default: equatorial · Choices for the current settings: Equatorial; Real polar.
- Angle (turns)
- Default: 0.125 · A fraction of a full turn, 0 ≤ φ < 1 (so φ = 0.5 means π radians).
- Apply Bob's correction
- Default: true · On or off.
Implementation notes
Alice q0 and Bob q1 first share Bell Phi+. For the agreed equatorial or real polar great circle, Alice's target-dependent local basis measurement produces one bit. Bob's fixed Z (equatorial) or Y (polar) correction for c0=1 prepares the target in either branch. Only Alice's gates depend on the target angle. This one-bit guarantee is restricted to these ensembles, not arbitrary known qubits. Corrections off leaves orthogonal branch states; without Alice's record Bob has I/2, so this does not signal without classical communication. Bob is left unmeasured: Classical bits shows Alice's record, not the output phase. Inspect Bob's state amplitudes or Bloch view. Angle is in turns (one turn is 2pi).
Default example: predictions & circuit facts
These values describe the unedited default setup. They are predictions and circuit properties, not measured results from your saved execution.
- Target amplitudes (0, 1)
- 0.7071+0.0000i, 0.5000+0.5000i
- Alice outcomes
- 0 / 1: 50% each
- Bob correction
- Z if c0=1
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.