Finding lowest Eigenvalue for given Two-Qubit Hamiltonian using VQE-like circuits
Hamiltonian (H): $$ H = \begin{vmatrix} 1 & 0 & 0 & 0\ 0 & 0 & -1 & 0\ 0 & -1 & 0 & 0\ 0 & 0 & 0 & 1 \end{vmatrix} $$
It works on variational principal which say if we have a Hamiltonian H with eigenstates and associated eigenvalues . Then the following relation holds: $$ H |\psi⟩ = \lambda|\psi⟩$$ where λ is energy value for given state |ψ⟩. for every different |ψ⟩ we can find its energy, but for only one |ψ⟩ there exist λ which is smallest of all, and we call that |ψ⟩ as groung state of the system. and VQE helps us to find groung state of any given system. to make process simpler it decompose Hamiltonians into Pauli-Matrices. Resourse
VQE can be sum up in three parts
- Decomposition
- Circuit
- Ansatz
- Initializing mesurement basis
- Measurement
(run the code for interactive eigenvalues graphs :) Preview:https://nbviewer.jupyter.org/github/pratik-ingle/VQE/blob/master/VQE.ipynb