Task-4: Find the lowest eigenvalue of the given matrix

• First I tried to find out eigenvalues and the one which is lowest among them for Hamiltonian classically
• I first tried to simulate VQE without noise
  • For that, I have first decomposed the hamiltonian in terms of a linear combination of the tensor products of Pauli operators.
  • Then I have applied the Variational method to get the minimum energy
  • Created a function so that we can measure on all three basis
  • I calculated the expectation values on Z basis and calculated the summation of them. This function I designed such that it will work for both Noiseless and Noise cases.
  • Applied to minimize function to find the opimal parameter with the method as COBYLA.
  • Plotted a graph to get a better understanding by first getting expectation values for different parameters ranging from -π to π.
  • Doing this, I was able to obtain the eigenvalue ≈ -0.9995.
• Next, I have tried to simulate VQE with noise.
  • First I tried to understand the Noise Model which constructs an approximate noise model, then passing the basis gates and coupling map. Also I tried to understand the qubit properties for the device_backend I used.
  • Created a similar function to summation with some minor changes
  • Repeating the penultimate step used in Noiseless simulation with again some minor changes
  • Again simulated graph but now the graph was looking different and the eigenvalue obtained was ≈ -0.7751.
  • I tried to overlay the graphs where I got good intuition of the effect of noise on the circuit created along with intuition of what would be the outcome achieved, if we run this on a real quantum computer.