Quantum Swap Test: State Overlap Estimator

State |ψ⟩ (register A)

Polar angle θ_A 60°

Azimuth φ_A 0°

State |φ⟩ (register B)

Polar angle θ_B 90°

Azimuth φ_B 40°

Swap-Test Shots

measured |0⟩measured |1⟩

True fidelity |⟨ψ|φ⟩|²

0.00

Predicted P(ancilla=0)

0.00

Estimated fidelity (shots)

—

Total shots fired

How it works

The swap test is the standard subroutine quantum machine learning algorithms use to estimate how similar two quantum states are — the quantum analogue of a distance/kernel function, used inside quantum k-means, quantum kNN and quantum kernel estimation.

Circuit:  |0⟩ ancilla — H — •SWAP• — H — measure
                          |ψ⟩ ——————×—————————
                          |φ⟩ ——————×—————————

P(ancilla = 0) = 1/2 + 1/2·|⟨ψ|φ⟩|²
P(ancilla = 1) = 1/2 − 1/2·|⟨ψ|φ⟩|²

An ancilla qubit is put into superposition with a Hadamard gate, then used to control a SWAP of registers A and B, then Hadamard'd again and measured. If |ψ⟩ = |φ⟩ exactly, the controlled-SWAP does nothing observable and the ancilla always reads |0⟩. If the two states are orthogonal, the ancilla reads |0⟩ or |1⟩ with equal 50/50 probability — the two extremes bracket every possible overlap.

θ, φ sliders — set each register's state on its Bloch sphere: |ψ⟩ = cos(θ/2)|0⟩ + e^iφsin(θ/2)|1⟩.
B = A — snaps register B onto register A (fidelity 1, ancilla always |0⟩).
B ⟂ A — snaps B to the antipodal point on the Bloch sphere (fidelity 0, a true orthogonal state).
Run 100 shots — fires 100 independent swap-test circuits using the true Born-rule probability above; no single shot ever reveals the state — only the accumulated statistics converge on the fidelity, exactly like a real quantum device.

This is why the swap test matters for QML: it lets an algorithm compare two data points encoded as quantum states — the core operation behind quantum distance-based classifiers and quantum clustering — without ever performing a full state tomography, which would need exponentially many measurements.

|ψ⟩ register A |φ⟩ register B ancilla qubit

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