Curriculum / Quantum Algorithms / Quantum Interference
Quantum Interference
Understand how interference powers quantum algorithms.
Quantum Interference
What Is Interference?
Interference is the phenomenon where two wave amplitudes combine, sometimes reinforcing each other, sometimes cancelling. It is why noise-cancelling headphones work, why soap bubbles are iridescent, and why quantum computers can compute.
In quantum mechanics, every possible path a system can take from one state to another has a complex-valued amplitude. When multiple paths lead to the same final state, their amplitudes add before the probability is computed. This is fundamentally different from classical probability, where probabilities add. Probabilities can only be positive, so separate paths always make an outcome more likely. Amplitudes can be negative, zero, or imaginary, so separate paths can cancel.
This is sometimes phrased as: "quantum mechanics is a theory of probability amplitudes that can be negative." It is the single most important fact about quantum computation.
Classical probability: P(A or B) = P(A) + P(B) − P(A and B) ≥ 0 always. No cancellation possible. Quantum amplitude: φ(A or B) = φ(A) + φ(B), where φ can be negative or complex. Two paths can cancel completely. This is why quantum computers can solve certain problems efficiently, they are engineered to make wrong-answer amplitudes cancel and right-answer amplitudes grow. Measurement extracts the result only at the end.
The H-H Circuit: Interference Made Explicit
The simplest quantum interference experiment is the double-Hadamard circuit. Starting from |0⟩:
After first H: State = (|0⟩ + |1⟩)/√2. Amplitude for |0⟩ is +1/√2; amplitude for |1⟩ is +1/√2.
After second H: Each state's amplitude is split equally and distributed to both outputs with a sign pattern:
| H contribution to |0⟩ output | H contribution to |1⟩ output | |
|---|---|---|
| From |0⟩ path (amplitude +1/√2) | +1/√2 × (+1/√2) = +1/2 | +1/√2 × (+1/√2) = +1/2 |
| From |1⟩ path (amplitude +1/√2) | +1/√2 × (+1/√2) = +1/2 | +1/√2 × (−1/√2) = −1/2 |
| Total amplitude | +1/2 + 1/2 = +1 | +1/2 − 1/2 = 0 |
This is the opening of the lesson. The full walkthrough, the interactive circuit, and the graded challenge continue inside myqubit.
How this lesson works
A guided reading lesson with interactive knowledge checks. Concepts are explained step by step with circuit diagrams and runnable examples, and you confirm understanding before moving on.
Part of: Quantum Algorithms
Learn the algorithms that make quantum computers powerful.
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