What do we learn from doing this? We learn a lot about the electronics states of the system. You can imagine if you have some system, say water or even let’s say something simpler and diatomic like hydrogen or nitrogen. We can imagine there is some equilibrium length that there should be distance apart. As it gets further apart, well they are going to stop having as much attractive interactions and then as it gets closer together start having repulsive interactions that push them apart. There will be some happy medium, but what’s determining those forces on the atoms, whether it is going to go closer or whether it is going to go further apart is the electronic states.
The electrons are really dictating where this equilibrium point is. With a quantum computer we can solve it at every single point, here, then here, then here, then here as it is going apart it understands where the equilibrium point is. Where is the electron causing these things would be attracted and repelled, so then it becomes stationary. To elaborate a little bit further, we are solving this electronic problem. We going to take this atom, we are going to ding it and hear the frequency response to the energy, but we’re going to solve it at every fixed nuclear distance apart.
This is called the Born-Oppenheimer Approximation, where we assume that the electrons will be so much faster then nuclei that we do not have to take into account, nuclei are effectively stationary. So nuclei are stationary at these two points, electrons are moving all around them, withizzing, and were trying to figure out what the energy of the electrons are. So for the two nuclei being this far apart we’re calculating the energy in the whole molecule and then we adjust the distance a little bit and we calculate the energy again. Exactly, and then this will map out an entire surface of configurations. Right, you should think what we doing is we’re actually fixing the nuclei.
And then we’re calculating with electrons, what the interactions with electrons and nuclei are, in order to understand electronic energy. And we can get the electronic energy at every single configuration, so this tells us whether it is going to be cis or it is going to be trans configuration of say an atom with some rotational degrees of freedom, and this will tell you which rotation direction is preferred because the interactions and you can also know this thing as quantitatively rather than qualitatively that you might have learned, inside perhaps a chemistry course or inside of some other solid state course or whatever course you have taken.
Why is it faster on a quantum computer? What is it about the quantum computer that makes this faster? The example I’d like to think of, when I think of why quantum computers are better at simulating quantum physics, is the example of test what are these things called, “crash dummies or car crash dummies.” Crash test dummies. Crash test dummies, exactly. When these crash test dummies, they put sensors all over them and they crash the car and they see what happened to them as they banged their head against the dashboard or whatever, but they don’t do any computation per se.
There is no computational model where you take F equals ma and compute what the forces are, no you just crash the thing and see what came out. The idea of classic computation is to take all these equations, F equals ma, H Psi equals E Psi, whatever equations that we have and then we solve them by going through the, some linear algebra or whatever technique you going to use, but with quantum computation we’re skipping all those things. It is much close to this analogy of the simulation. It is really quantum simulation as much as quantum computation. So the computational part is just to map, one system to the other.
But the simulation part is just, let it run and see what happens. So this is the same idea with the crash test dummies that is the analogy I’d like to think of. So in mechanical systems and electronic systems, for example if you are trying to figure out how a bridge resonates or vibrates, you can measure some of the characteristics of that and then map those characteristics to an electrical circuit using resistors and capacitors and inductors and then just let the circuit run for a little while and you can see what the resonant frequencies are and things like that about the actual behavior, is this same kind of thing?
I would say it is the same kind of thing, but I would think a closer analogy would be that you had a large scale bridge and you built a tiny one that you can really have little tiny soldiers marching across and see how it vibrates. Because in the example you gave is that you are really taking that’s an electronic circuit and mapping to a physical system, but in our case we are taking a quantum system and mapping it to another quantum system. Like we did not even change the formulation of the system.
We just use qubits to describe the quantum system and then map it on some discrete set that we can describe some finite space, finite time, and whatever allocations you have, the computational resource constraints that you have. Nonetheless, it’s the same thing as building a small model of whatever system you are interested in. Maybe not even smaller, but just a model of it and you are going be a controllable model. Now we can control this model, we might be able to go very slowly to a process that might be very complicated and it occurs very quickly in nature that we can now step through it, almost like debugging. So how hard is this going to be on quantum computers?
Is this going to be a computation that takes thousands and thousands of qubits and it is going to take years to run or is it something we can do with 3 qubits and just a little bit of effort? Okay that is a great question. To answer, let me answer in two parts. First spatially and temporally. If you think spatially, the number of qubits you have is going to dictate the best description of the system you can make. So you have to discretize to some level. If you think of a perfect circle, when you put it on a digital screen it becomes discretized at some level.
Maybe you can’t see it, maybe you can, depends the level of pixels and you could think the number of qubits you have is how well you are going to be able to copy an image or simulate the system effectively. So it is not necessarily the case that if we want to simulate a molecule with 5 atoms and we can use 5 qubits.
Right, that is not necessarily the case and in fact what you can do is imagine, that you brutalize the system and kept pushing it down further and further until it fit into the size that you had versus having a very-very large system and you can have very accurate description of it inside there is very large number of qubits states or quantum register that you have. Like simulating weather where simulating you are simulating one cube of the atmosphere and maybe you make the cubes smaller and smaller that gets you a better weather prediction. Coarse graining versus not coarse graining. And how fine is your grid and how fine is your mesh that you describe in a system with.
That is exactly right, that is a perfect example. If you want to describe the weather if you could take voxels of 1 meter squared where you can have a lot of them versus taking 100 kilometers at the time. So that is the spatial aspect of it. Right. Now the important part of the temporal aspect and this is where everything comes to the head and why quantum computation really is an important tool going forward in terms of quantum simulation. Is that there are theorems that the amount of time it takes to simulate a quantum system is linearly proportional to the amount of time that quantum system takes to simulate itself. Okay. So the point is that you are just scaling.
So if it takes you twice or if you want to simulate for twice as long then it really should only take you twice as many gates. So this where quantum computation really gets this effective speed up. Because you are really simulating a system rather than trying to solve all these equations. When you do it in a classic computer, regardless of the discretization level the computational time it takes is going to be much longer and it is going to be dependent on what algorithm you are using on how you approach this algorithm. Whether you use meanfield and all these different techniques. Not perfect but…
For the kinds of problems that you are interested in or the kinds of problems that people are already working on how many qubits and how hard a computation is this? One of the first quantum simulations, that I was happy to be a part of, was simulating hydrogen. We used two qubits to simulate hydrogen, which is a very small system. You can actually do it by hand oddly enough. This was just our initial simulation of hydrogen, but again with hydrogen you can keep adding more qubits to describe this same system and get a better description of that system, so really what I like to think of is as quantum computers get bigger are going to get better simulation output.
So the discretization error will go down. Sounds good. James thank you for being here with us in the course and good luck in all your research. Thanks. Hopefully that gave you something to edit. I think so.