Phase estimation algorithm

Phase estimation algorithm
test

PEA algorithm

Phase estimation algorithm or PEA is used for quantum chemistry application, for estimating the energy eigenvalues of a molecular Hamiltonian. The PEA can be used to estimate the value of the phase witch allows determining the corresponding eigenvalues of the Hamiltonian.
Phase estimation is a discretization of von Neumann’s prescription to measure a Hermitian observable.  A measurement is made of a simple observable e.g. the location. We make a convenient observation from quantum mechanics, it’s known that a momentum operator p generates shifts for single particles.
particle_shift
We have a controlled U or Unitary operator U = e^{i\theta}. That is, if we apply the unitary to some wave packet, then this wave packet will be shifted by in the positive direction.
Furthermore, the target register is prepared in some state phi.

In above circuit, the target register is prepared in some state Phi. A controlled version of U is a unitary operator acting on the system control and target, where control is a single qubit and target is a register if n qubits. Controlled U applies U to the target register if the control bit is  \ket{1}. The control bit gets mapped from \alpha + \beta to, while the target register remains in the state phi. Thus we can describe that the action of controlled-U on the composite system control + target with a single qubit phase shift gate acting on the control bit.

Phase kick back

PEA estimates the phase \phi in the eigenvalue  e^{i\theta} of a unitary transformation. A controlled U can be used for a phase kick back circuit.  Quantum circuit for phase estimation algorithm.
phase_kickback
\frac{\ket{0} + \ket {1}}{\sqrt 2} \bigotimes \ket{\psi} = \frac{\ket{0} }{\sqrt 2}\bigotimes\ket{\psi} + \frac{\ket{1} }{\sqrt 2}\bigotimes\ket{\psi}
\mapsto \frac{\ket{0} }{\sqrt 2}\bigotimes\ket{\psi} + \frac{\ket{1} }{\sqrt 2}\bigotimes U\ket{\psi}
= \frac{\ket{0} + e^{2\pi i\phi}\ket{1}}{\sqrt 2}\bigotimes\ket{\psi}
 

Hadamard

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