Quantum phase estimation algorithm
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 controlled-U operations.
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 input consists of two registers (namely, two parts): the upper qubits comprise the first register, and the lower qubits are the second register.
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
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
- Chuang, Michael A. Nielsen & Isaac L. (2001). Quantum computation and quantum information (Repr. ed.). Cambridge [u.a.]: Cambridge Univ. Press. ISBN 978-0521635035.
- Benenti, Guiliano; Casati, Giulio; Strini, Giuliano (2004). Principles of quantum computation and information (Reprinted. ed.). New Jersey [u.a.]: World Scientific. ISBN 978-9812388582.
- 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). arXiv:quant-ph/9708016. Bibcode:1998RSPSA.454..339C. doi:10.1098/rspa.1998.0164.