swap test

In plain terms

Two devices each hand you a quantum state, and you want to know whether they agree. Measuring either state disturbs it and reveals only a little about it, and comparing the two lists of amplitudes directly isn't an operation nature offers.

The swap test threads one extra qubit, called an ancilla, through both states. Put the ancilla into a superposition of its two readings, and in the branch where it reads one swap the two states while in the other branch you leave them alone, then close the superposition so the two branches interfere, and read the ancilla. If the two states are identical, the swap changed nothing, and the ancilla reads zero every time, and if the states are orthogonal, meaning perfectly distinguishable, the ancilla is a fair coin. In between, the bias of the coin measures the overlap of the two states, the standard gauge of their resemblance, which runs from zero for perfectly distinguishable states to one for identical ones.

Repeat as often as you like and the overlap is estimated to any accuracy, while neither state is ever identified. It's the standard likeness-meter of quantum information.

In CSD

The swap test is the witness that opens the corpus's ancilla-interferometry family, the algorithms that read a quantity off the interference of a single helper qubit, beside the oracle family of Deutsch-Jozsa, Bernstein-Vazirani and Simon. The probability that the ancilla reads zero is proved to be exactly one half plus half the squared overlap of the two states, with the identical-state and orthogonal-state readings as corollaries.

The standing disclaimer applies, as the algorithms consume the programme's unitary- evolution pillar and don't support the foundations.

Mathematically

H on the ancilla, controlled swap, H again, on |0> tensor psi tensor phi, and the ancilla-zero marginal is (1/4) times the squared norm of psi tensor phi + phi tensor psi, so the cross term <psi tensor phi, phi tensor psi> = |<psi, phi>|^2 closes it, giving P(0) = (1 + |<psi, phi>|^2)/2 exactly (swap_test_prob).

P(0) = 1 iff the states coincide up to phase, P(0) = 1/2 iff they're orthogonal, and the estimator is quadratic in the overlap, which is why it reveals magnitude and not phase.

Related entries

• Hadamard testThe Hadamard test turns a matrix element, a complex number buried inside an operator, into the bias of a single coin.
• amplitude estimationPoint phase estimation at the rotation inside amplitude amplification and the success probability itself becomes measurable, quadratically faster than sampling could ever learn it.
• Hong-Ou-Mandel effectWhen two identical photons meet at a beam splitter they always leave together and never one through each exit, so their indistinguishability shows up as a hole in the data.
• Fisher informationThere's a floor under every error bar, classical or quantum, and Fisher information is what sets it.
• Grover's algorithmGrover's algorithm finds one marked item in an unsorted list in a number of steps proportional to the square root of the list size, and a separate theorem shows that no quantum algorithm can do better.