Quantum Computing for Natural Sciences (with IBM Quantum)

1. 00:00Welcome back to the course on quantum computing for natural sciences.
2. 00:04In the previous lectures, we have seen how we can map an electronic structure problem onto the qubit space where we can then
3. 00:10use quantum algorithms such as the variational quantum Eigensolver to find the ground state solution of an electronic structure
4. 00:16problem.
5. 00:17In this lecture, we will look at embedding methods, a technique that will allow us to reduce the size of our problems to
6. 00:24fit them on quantum computers that we have available today.
7. 00:29In the past research on quantum computing has been done using small scale molecules such as a single water molecule, nitrogen
8. 00:36oxygen as shown here on the left.
9. 00:39However, of relevance for chemical industry, we actually have much larger scale problems such as shown here on the right
10. 00:46hand side where we have iron sulfur complexes, a protein or even periodic systems embedding techniques allow us to reduce
11. 00:54our system problem sizes to a scale that will allow us to treat at least intermediately sized problems such as shown here
12. 01:01in the center on quantum computers
13. 01:03Today examples are the butanenitrile (butyronitrile) shown at the top shown, pyridine on the right or an iron porphyrin model shown here at
14. 01:11the bottom.
15. 01:13Let us take a look at how we can actually achieve such an embedding technique.
16. 01:17One way is the so-called active space embedding.
17. 01:20The idea being that the relevant chemistry really only happens in a small part of our molecule.
18. 01:27A key example is hemoglobin, a protein that is relevant to our understanding of biological organisms um as they exist
19. 01:37in nature today, a subset of this hemoglobin is the iron porphyrin model.
20. 01:44Uh This is at the core of this iron of the hemoglobin.
21. 01:47And what we can see is that there are already methods out there that use classical computing to embed a high level description
22. 01:55of iron porphyrin into hemoglobin.
23. 01:57However, on quantum computers, we cannot even fit the iron porphyrin model today.
24. 02:02That's why we need to further reduce our system size.
25. 02:05And one way of doing that is by using an active space.
26. 02:08On the left hand side here, I'm showing the entire molecular orbitals of iron porphyrin pictographicly.
27. 02:15The orange box shows that we have orbitals that are occupied at the bottom and virtual at the top that describe our system
28. 02:22classically.
29. 02:23And we can now set select an active subset in blue that we will actually put onto the quantum computer.
30. 02:30The key part here is that we have both occupied and virtual orbitals as part of our active space such that we can actually
31. 02:37include excitations in our slater determinants.
32. 02:42Let's take a look at how this works mathematically, this is the electronic structure Hamiltonian, which we've already seen
33. 02:49back in lecture two.
34. 02:50Here we have the one body and two body interaction intervals, H P Q and H P Q R S.
35. 02:56And the key to the active space embedding is now that we are only looking at a reduced size problem where we are replacing
36. 03:02our one body terms with the inactive Fock operator.
37. 03:06This inactive Fock operator includes both the one body interactions, which are the electronic, the electronic kinetic part as
38. 03:14well as the electron and nuclei attraction force.
39. 03:17But also the screen two body interactions by screen
40. 03:20I mean that we are summing over the inactive inactive orbitals or inactive electrons.
41. 03:27Here we have the electron electron repulsion term the Coulomb interaction of our inactive electrons with the ones in the active
42. 03:33space.
43. 03:34And we have the exchange interaction of our inactive system with the active one.
44. 03:39Let's take a look at how we can actually realize this in code.
45. 03:42We've already used Qiskit nature in previous lectures.
46. 03:45And this time we will be using the PySCFDriver and the active space transformer, the PySCFDriver
47. 03:51Here is being set up for a simple N2 molecule.
48. 03:54This is a piece of code which will use PySCF a classical computational code for solve quantum chemistry problems.
49. 04:02And we will use that to find the Hartree-Fock solution of our system as our starting point, when we run this driver, we
50. 04:09will get out an electronic structure problem here
51. 04:12We can see that this electronic structure problem for N2 in the given basis has seven alpha spin and seven beta spin electrons
52. 04:19distributed over 28 spatial orbitals.
53. 04:23This is exceeding the size that we can fit onto a quantum computer by far.
54. 04:28So now we can use an active space transformer to really scale this down to a system that we can treat here
55. 04:35We are picking four electrons in four spatial orbitals as our target size.
56. 04:40And indeed applying this transformer to our problem
57. 04:42We now get a new electronic structure problem using that transformation that I've shown on the previous slide to get a two
58. 04:49alpha spin and two beta spin electrons in four spatial orbitals sized problem.
59. 04:54We can now use the techniques that Julian showed in the previous lecture to solve the ground to find the ground state solution
60. 05:01of this electronic structure problem.
61. 05:03I'm not going to repeat that I will jump straight to the result.
62. 05:06And here we can see that we find the ground state energy consisting of multiple parts, the actually computed part the active
63. 05:15space extracted energy, which is the inactive energy component together these two sum to the electron electronic ground state
64. 05:22And then adding on top of the nuclear repulsion energy, we arrive at the total ground set energy of our system.
65. 05:28This is very simple as you can see, and you will find the notebook attached to the additional resources of the lecture.
66. 05:36Let's also take a look at how this is actually used in research.
67. 05:39Back in 2021 we published a paper introducing these methods for quantum computing.
68. 05:44And on the left hand side, I'm showing some examples as we can recover more and more of the electronic correlation that is
69. 05:51the gap between the black Hartree-Fock energy and the red target energy.
70. 05:57And as you can see, as we scale up the active space, as we move towards the right on the X axis, we can recover
71. 06:03more and more of that energetic contributions.
72. 06:08We also in that paper introduce the methods how we can embed into density functional theory or D F T another low scaling
73. 06:17classical method besides Hartree-Fock this allows you to scale even further to larger systems
74. 06:23But again, here, I'm simply showing the results for a single water molecule.
75. 06:28These are the dissociation energies.
76. 06:30Um And they show that we can recover the qualitatively correct association using small active spaces regardless of whether
77. 06:37we start from Hartree-Fock on the right panel or D F T on the left panel.
78. 06:41I encourage you to take a look at the paper to dive more deeply into how we can use active space embedding techniques to
79. 06:48scale our quantum computational calculations
80. 06:51Today
81. 06:53Next week, we will step outside of the ground state realm and look at excited set energies as well as quantum time evolutions

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