In quantum computing, the quantum phase estimation algorithm (also referred to as quantum eigenvalue estimation algorithm), is a quantum algorithm to estimate the phase (or eigenvalue) of an eigenvector of a unitary operator. More precisely, given a unitary matrix and a quantum state such that , the algorithm estimates the value of with high probability within additive error , using qubits (without counting the ones used to encode the eigenvector state) and controlled-U operations. The algorithm was initially introduced by Alexei Kitaev in 1995.[1][2]: 246 

Phase estimation is frequently used as a subroutine in other quantum algorithms, such as Shor's algorithm[2]: 131  and the quantum algorithm for linear systems of equations.

The problem

Let U be a unitary operator that operates on m qubits with an eigenvector such that .

We would like to find the eigenvalue of , which in this case is equivalent to estimating the phase , to a finite level of precision. We can write the eigenvalue in the form because U is a unitary operator over a complex vector space, so its eigenvalues must be complex numbers with absolute value 1.

The algorithm

Quantum phase estimation circuit
Quantum phase estimation circuit


The input consists of two registers (namely, two parts): the upper qubits comprise the first register, and the lower qubits are the second register.

Create superposition

The initial state of the system is:

After applying n-bit Hadamard gate operation on the first register, the state becomes:


Apply controlled unitary operations

Let be a unitary operator with eigenvector such that thus by exponentiation by squaring,


is a controlled-U gate which applies the unitary operator on the second register only if its corresponding control bit (from the first register) is .

Assuming for the sake of clarity that the controlled gates are applied sequentially, after applying to the qubit of the first register and the second register, the state becomes

where we use:

After applying all the remaining controlled operations with as seen in the figure, the state of the first register can be described as

where denotes the binary representation of , i.e. it's the computational basis, and the state of the second register is left physically unchanged at .

Apply inverse quantum Fourier transform

Applying inverse quantum Fourier transform on


The state of both registers together is

Phase approximation representation

We can approximate the value of by rounding to the nearest integer. This means that where is the nearest integer to and the difference satisfies .

We can now write the state of the first and second register in the following way:


Performing a measurement in the computational basis on the first register yields the result with probability

For the approximation is precise, thus and In this case, we always measure the accurate value of the phase.[3]: 157 [4]: 347  The state of the system after the measurement is .[2]: 223 

For since the algorithm yields the correct result with probability . We prove this as follows:[3]: 157 [4]: 348 

This result shows that we will measure the best n-bit estimate of with high probability. By increasing the number of qubits by and ignoring those last qubits we can increase the probability to .[4]

See also


  1. ^ Kitaev, A. Yu (1995-11-20). "Quantum measurements and the Abelian Stabilizer Problem". arXiv:quant-ph/9511026.
  2. ^ a b c Nielsen, Michael A. & Isaac L. Chuang (2001). Quantum computation and quantum information (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 978-0521635035.
  3. ^ a b Benenti, Guiliano; Casati, Giulio; Strini, Giuliano (2004). Principles of quantum computation and information (Reprinted. ed.). New Jersey [u.a.]: World Scientific. ISBN 978-9812388582.
  4. ^ a b c Cleve, R.; Ekert, A.; Macchiavello, C.; Mosca, M. (8 January 1998). "Quantum algorithms revisited". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 454 (1969): 339–354. arXiv:quant-ph/9708016. Bibcode:1998RSPSA.454..339C. doi:10.1098/rspa.1998.0164. S2CID 16128238.