Day: December 8, 2025

Kitaev Algorithm for Quantum Phase Estimation

In this post, we introduce the quantum phase estimation based on quantum Fourier transform. There is another way to implement quantum phase estimation via an interferometric scheme, usually called Kitaev algorithm.

As before, we assume that we can prepare an eigenstate |\psi\rangle of a unitary operator U,

U|\psi\rangle = e^{2\pi i \theta} |\psi\rangle ,

where we take \theta \in [-1/2,1/2) for convenience. Kitaev’s idea is to encode the phase information into the amplitudes of certain superposition states and extract \theta using the interferometric scheme.

Fig 1: interferometric scheme for cosine part.

The quantum circuit is shown in Figure 1.

Acting with a Hadamard gate on the control qubit gives

\frac{1}{\sqrt{2}}(|0\rangle|\psi\rangle + |1\rangle|\psi\rangle).

Applying the controlled-U gate yields

\frac{1}{\sqrt{2}}(|0\rangle|\psi\rangle + e^{2\pi i\theta}|1\rangle|\psi\rangle),

so the phase is now encoded in the relative amplitude. A final Hadamard gate transforms the state to

|\Phi_f^{c}\rangle = \frac{1}{\sqrt{2}}(|+\rangle|\psi\rangle + e^{2\pi i\theta}|-\rangle|\psi\rangle)

= \frac{1+e^{2\pi i \theta}}{2}|0\rangle|\psi\rangle + \frac{1-e^{2\pi i\theta}}{2}|1\rangle|\psi\rangle

Thus the measurement probabilities are

p(0) = \left|\frac{1+e^{2\pi i\theta}}{2}\right|^2 = \frac{1+\cos(2\pi\theta)}{2}

p(1) = \left|\frac{1-e^{2\pi i\theta}}{2}\right|^2= \frac{1-\cos(2\pi\theta)}{2}

These measurements allow us to extract the cosine of the phase,

\theta = \frac{1}{2\pi}\cos^{-1}(2p(0)-1)

or equivalently,

\theta = \frac{1}{2\pi}\cos^{-1}(1-2p(1))

However, we know that \cos(2\pi \theta) cannot distinguish between
\theta and -\theta.
It is therefore natural to incorporate the sine component \sin(2\pi\theta) to resolve this ambiguity.
By first determining the sign of \theta from the sine measurement, we can then output the final estimate as

\theta = \pm \frac{1}{2\pi}\cos^{-1}(2p(0)-1),

where the sign is chosen according to whether \sin(2\pi\theta) is positive or negative.

Fig 2. interferometric scheme for sine part.

To obtain the sine component, we slightly modify the circuit by inserting a phase gate S on the control qubit prior to the controlled-U operation, see Figure 2.

A parallel computation yields

|\Phi_f^{s}\rangle = \frac{1+ie^{2\pi i\theta}}{2}|0\rangle|\psi\rangle+\frac{1-ie^{2\pi i\theta}}{2}|1\rangle|\psi\rangle.

The measurement probabilities are now

p(0) = \left|\frac{1+ie^{2\pi i\theta}}{2}\right|^2= \frac{1-\sin(2\pi\theta)}{2}, \quad p(1) = \left|\frac{1-ie^{2\pi i\theta}}{2}\right|^2= \frac{1+\sin(2\pi\theta)}{2}

Therefore,

\theta = \frac{1}{2\pi}\sin^{-1}(1-2p(0))

or equivalently,

\theta = \frac{1}{2\pi}\sin^{-1}(2p(1)-1)

with \sin^{-1} taking values in [-\pi,\pi).

Combining the estimates of \cos(2\pi\theta) and \sin(2\pi\theta) allows us to reconstruct \theta uniquely in the range [-1/2,1/2).