Phase Estimation Algorithm


The phase estimation algorithm is a quantum subroutine useful for finding the eigenvalue corresponding to an eigenvector \(u\) of some unitary operator. It is the starting point for many other algorithms and relies on the inverse quantum Fourier transform. More details can be found in references [1].


First, connect to the QVM.

import pyquil.api as api

qvm = api.QVMConnection()

Now we encode a phase into the unitary operator U.

import numpy as np

phase = 0.75
phase_factor = np.exp(1.0j * 2 * np.pi * phase)
U = np.array([[phase_factor, 0],
              [0, -1*phase_factor]])

Then, we feed this operator into the phase_estimation module. Here, we ask for 4 bits of precision.

from grove.alpha.phaseestimation.phase_estimation import phase_estimation

precision = 4
p = phase_estimation(U, precision)

Now, we run the program and check our output.

output =, range(precision))
wavefunction = qvm.wavefunction(p)


This should print the following:

[[0, 0, 1, 1]]

Note that .75, written as a binary fraction of precision 4, is 0.1100. Thus, we have recovered the phase encoded into our unitary operator.

Source Code Docs

Here you can find documentation for the different submodules in phaseestimation.



Make a one-qubit-controlled version of a matrix.

Parameters:m – (numpy.ndarray) A matrix.
Returns:A controlled version of that matrix.
grove.alpha.phaseestimation.phase_estimation.phase_estimation(U, accuracy, reg_offset=0)

Generate a circuit for quantum phase estimation.

  • U – (numpy.ndarray) A unitary matrix.
  • accuracy – (int) Number of bits of accuracy desired.
  • reg_offset – (int) Where to start writing measurements (default 0).

A Quil program to perform phase estimation.


[1]Nielsen, Michael A., and Isaac L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2010.