Quantum Computing for Natural Sciences (with IBM Quantum)

1. 00:00Hello and welcome back to the course on quantum computing for Natural Sciences.
2. 00:05In the last lecture, you have seen two quantum algorithms, the quantum phase estimation and the variational quantum Eigen
3. 00:11solver to solve for the ground state of the time independent Schrödinger equation.
4. 00:16In the context of quantum chemistry.
5. 00:19Today, you are going to see how we implement the version of quantum Eigensolver with the software package Qiskit developed
6. 00:27by I B M.
7. 00:30In the following, you are going to see how to how to code to implement each part of the VQE algorithm, which means how
8. 00:39to map the molecule to the qubit Hamiltonian, how to define the Ansatz and how to define the optimization routine.
9. 00:47And finally how to set and run the VQE algorithm.
10. 00:52As the first step we collect information about our molecule of interest.
11. 00:57In our case, it's the lithium hydride molecule the lithium and the hydrogen atom set 1.3 angstrom apart in the Qiskit code
12. 01:07we can use the PySCFDriver giving it the information about the molecule and the basis within which we want to work
13. 01:15In our case, it is this sto3g basis.
14. 01:20In the next step, we then define the Hamiltonian in its second quantized formulation.
15. 01:27For this, we run the driver to get the electronic structure problem from the electronic structure problem
16. 01:34We can then extract this Hamiltonian in its second quantization formulation.
17. 01:40In the next step, we then map to Hamiltonian to the corresponding qubit Hamiltonian.
18. 01:47For this, we are using the parity mapper, it uses the number of particles which means the number of electrons as an argument
19. 01:55Then we can apply the parity mapper to the Hamiltonian extracted in the previous step.
20. 02:04Having the qubit Hamiltonian ready.
21. 02:06We can now go to the next step and define the parameterized Ansatz.
22. 02:11The parameterized Ansatz consists of two parts first, the fixed initial state, for which we use the Hartree-Fock state.
23. 02:19And the second part, the parameterized circuit for which we use the unitary couple cluster with single and double excitations
24. 02:28That was introduced in the previous lecture.
25. 02:31Both of them need the number of spatial orbitals, which is the number of orbitals on which the problem is defined.
26. 02:38And the number of particles as an input argument.
27. 02:41Additionally, they also need to know about the mapper in order to correctly map the initial state and the excitations in the
28. 02:49UCCSD Ansatz to the corresponding circuits.
29. 02:55The last part is to initialize the optimization routine.
30. 02:59In our case, we are using the COBYLA optimizer with a maximum number of iterations set to 2500.
31. 03:07This is ensures that after 2500 iterations and the optimizer still has not converged, it still stops.
32. 03:18Then we can finally set up the VQE algorithm.
33. 03:22First, we set up the environment in which the VQE is executed.
34. 03:27Since we are executing the VQE locally, we can straightforwardly do this by calling the Qiskit run time primitive estimator
35. 03:36If you would like to execute the VQE algorithm on real hardware, we would have to specify this at this point, then we can
36. 03:44set up the VQE solver by passing the estimator the parameterized Ansatz and the optimizer to the VQE class,
37. 03:53then we can call the function compute minimum eigenvalue giving us argument to qubit Hamiltonian which starts the VQE
38. 04:02optimization.
39. 04:04After the optimization has converged or reached a number of maximal iterations, it returns a minimum a minimum eigensolver
40. 04:15result.
41. 04:17This result can then be transformed into an electronic structure result by passing the result to the interpret function of
42. 04:26the problem.
43. 04:28The final VQE energy can then be extracted from the results by calling the total energies variable
44. 04:37on the left
45. 04:38You can see the results that were obtained by the presented code.
46. 04:43The VQE energe is -7.7 Ha which is very close to the exact energy which for such a small problem
47. 04:54can be obtained by exactly diagnosing the Hamiltonian
48. 04:59below
49. 04:59You can also see a prototypical energy evolution.
50. 05:03During the VQE optimization, we start at a higher energy.
51. 05:08And after initial exploration phase in the parameter space, we converge to the exact ground state energy.
52. 05:18The only difference between the code that is used to create these results
53. 05:24And the present code is that we used, we mapped the problem to a smaller problem size using active space transformation.
54. 05:35This is something you are going to learn about in the next lecture to present the code and the steps necessary to reproduce
55. 05:42the results are going to be available in the additional resources of the course.
56. 05:50This code and additional code examples can also be found in the tutorials of Qiskit.

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