The Hadamard Test

Estimate the real (or imaginary) part of ⟨ψ|U|ψ⟩ using one ancilla qubit — a building block for Krylov methods and overlap estimation. It measures a complex number you cannot observe directly using simple interference on a single qubit.

The problem it solves

Many quantum algorithms need one number: the overlap ⟨ψ|U|ψ⟩ of a state |ψ⟩ with itself after a unitary U has acted on it. This is a complex number that captures how much of |ψ⟩ the operator U maps back onto |ψ⟩. It is generally a complex number that cannot be read out directly by one ordinary computational-basis measurement; although its real and imaginary parts can each be written formally as a Hermitian-operator expectation, the Hadamard test is the standard way to estimate them using controlled-U.

Re⟨U⟩ = ⟨(U+U†)/2⟩, Im⟨U⟩ = ⟨(U−U†)/(2i)⟩

The Hadamard test sidesteps this with a single ancilla qubit. It converts the hidden complex overlap into a statistical bias in a series of simple measurements of that ancilla, letting you estimate its real and imaginary parts separately.

The core idea — interference on one qubit

The ancilla is an interferometer. The first Hadamard puts it in an equal superposition of |0⟩ and |1⟩. A controlled U then applies U to |ψ⟩ only on the branch where the ancilla is |1⟩. This creates a relative phase between the two ancilla branches that carries the information in ⟨ψ|U|ψ⟩. The second Hadamard makes the two branches interfere, translating that phase difference into a measurable imbalance between the probabilities of |0⟩ and |1⟩.

Why P(0) minus P(1) gives the real part

After the interference sequence, the probabilities of measuring the ancilla as 0 or 1 are

P(0) = ½(1 + Re⟨ψ|U|ψ⟩), P(1) = ½(1 − Re⟨ψ|U|ψ⟩)

Subtracting the two, the constant and the ½ factor cancel, leaving cleanly

P(0) − P(1) = Re⟨ψ|U|ψ⟩

That is why assigning +1 to outcome 0 and −1 to outcome 1 and averaging yields the real part. To get the imaginary part, insert an S† gate on the ancilla before the controlled U: it rotates the relative phase between the branches by 90°, moving what was the imaginary part into the real-part slot, so the same readout now returns Im⟨ψ|U|ψ⟩.

Under this page's convention — controlled-U acting on the |1⟩ branch of the ancilla, read out as P(0) − P(1) — inserting S† gives +Im⟨ψ|U|ψ⟩; inserting S gives the opposite sign.

What it is — and isn't — good for

• It needs just one ancilla. However many qubits |ψ⟩ occupies, the overhead is a single ancilla plus one controlled U — a very simple, implementable structure.
• It needs a controlled U. You must be able to apply U in controlled form, which is not cheap for every operator; this is often the main source of circuit depth.
• It is statistical. You get an estimate: driving the error down to ε takes about O(1/ε²) samples, the inherent scaling of sampled measurement.
• It is a component of larger methods. Overlaps of the form ⟨ψ_i|H|ψ_j⟩ are exactly what Krylov-subspace methods and quantum eigensolvers need; the Hadamard test is how you measure them on hardware.

Circuit

import qalgora

@qalgora.kernel
def hadamard_test(imag: bool):
    anc = qalgora.qubit()
    q = qalgora.qvector(2)
    # prepare |psi> on q ...
    h(anc)
    if imag:
        s.adj(anc)               # imaginary part variant
    # controlled-U on q, conditioned on anc ...
    h(anc)
    mz(anc)

counts = qalgora.sample(hadamard_test, False)
# Re<psi|U|psi> = P(0) - P(1)

Seeing it in code

Read the kernel against the theory above. anc is the ancilla acting as the interferometer, and q carries the state |ψ⟩ (its preparation is left as a commented placeholder). The first h(anc) puts the ancilla in an equal superposition. When imag is true, s.adj(anc) applies the S† gate that switches the readout from the real to the imaginary part. The commented controlled U applies U to |ψ⟩ only on the |1⟩ branch of the ancilla, writing in the relative phase. The second h(anc) completes the interference and mz(anc) measures the ancilla. As the comment shows, processing the samples as P(0) − P(1) yields Re⟨ψ|U|ψ⟩.

This code is a circuit skeleton; only after you fill in the preparation of |ψ⟩ and the controlled-U does the sampled result correspond to a concrete ⟨ψ|U|ψ⟩.