Equal superposition of every single-excitation basis state. Entanglement survives the loss of one qubit.
For three qubits, |W⟩ = (|001⟩ + |010⟩ + |100⟩)/√3. Exactly one qubit is |1⟩ in every term. Discarding any qubit leaves a mixed entangled state of the other two, which is why W is used in some network and memory protocols.
The circuit seeds |1⟩ on the last wire, then applies F-gates (RY, CZ, RY) that split amplitude with θ = arccos(√(1/(n-k+1))), then a CX cascade that spreads the excitation.
How it works
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1
Seed
X 2 puts the excitation on q2: |100⟩.
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2
F-gates
Each F-gate is RY(-θ), CZ, RY(+θ). The first splits 1/√3 onto q1; the second splits 1/√2 of the remainder onto q0.
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3
Spread
CX [1,2] then CX [0,1] move the excitation into the symmetric W superposition.
X 2 RY 1 -0.195913 CZ [2,1] RY 1 0.195913 RY 0 -0.250000 CZ [1,0] RY 0 0.250000 CX [1,2] CX [0,1]
Function form: W(0..2)
State. |W⟩ = (|001⟩ + |010⟩ + |100⟩)/√3
P(001) = P(010) = P(100) = 1/3. Never 000, 011, 101, 110, or 111.
| Basis | Amplitude | Probability |
|---|---|---|
|001⟩ |
1/√3 |
|
|010⟩ |
1/√3 |
|
|100⟩ |
1/√3 |
Notes
- Angles are Qubi π-multiples: RY θ means a rotation of θπ.
- arccos(√(1/3))/π ≈ 0.195913. arccos(√(1/2))/π = 1/4.
Change parameters
The write-up above is for the default circuit. Use this control to generate other variants and load them in the simulator.