Quantum Interference in Detail

Interference is the constructive and destructive combination of probability amplitudes, and it is the resource behind quantum speedups.

Amplitudes add before they are squared

Classical probabilities of alternative paths simply add. Quantum amplitudes add first, and only the total is squared by the Born rule. If an outcome can be reached by two indistinguishable paths with amplitudes a and b, its probability is |a + b|^2 = |a|^2 + |b|^2 + 2 Re(a* b). The final cross term is interference: it can raise the probability above the classical sum (constructive) or cancel it entirely (destructive).

The double slit as the canonical case

With both slits open and no which-path information, amplitudes from each slit superpose and produce fringes. If any record of the path exists, even in principle, the paths become distinguishable, the cross term drops, and the pattern washes out into the classical sum. Interference thus requires coherence: a definite phase relationship that is not leaked to the environment. Decoherence is precisely the loss of this cross term.

Interference in algorithms

Quantum algorithms are engineered interference. A computation spreads amplitude across many basis states, applies phases that encode the problem, and then interferes them so that wrong answers cancel and right answers reinforce. Grover's search rotates amplitude toward the marked item through repeated reflections; the quantum Fourier transform converts encoded phases into measurable amplitudes. Without destructive interference removing the unwanted branches, sampling the final state would give no advantage over a random guess.

import numpy as np
H = np.array([[1,1],[1,-1]])/np.sqrt(2)
psi = np.array([1,0])          # |0>
psi = H @ psi                  # superposition
psi = H @ psi                  # interfere back
print(np.round(np.abs(psi)**2, 3))  # [1, 0] -> back to |0>

The two Hadamards illustrate the mechanism: the first creates equal amplitudes on |0> and |1>, the second makes the two paths to |1> cancel and the two paths to |0> reinforce, deterministically restoring |0>. This is destructive interference in its simplest form.

Phase is the control knob

Because only relative phases enter the cross term, controlling phase is controlling interference. Gates that impart problem-dependent phases, followed by a basis change that turns phase differences into amplitude differences, are the universal template. Preserving coherence long enough for the intended interference to complete is the central engineering demand on quantum hardware.