Okay, the press release here has dumbified the explanation to the point that several commenters are confused as to what it's trying to do, getting hung up on the H₄. The underlying chemistry is somewhat beyond me, and I have not an ACS subscription to speed-read the related research, and this is at the limits of what Wikipedia can describe, so I may be somewhat wrong, but I'm still going to try my best to explain the science being don here nonetheless.
Photon goes bonk on a molecule and it knocks an electron free (we can turn this electron into power in solar cells!). Sometimes, however, this electron knocks a second electron free, which is the "singlet fission" process being described (this means we get more power in our solar cells). We want to model this process in various molecules to be able to make better molecules for solar cells.
There's a problem... this is quantum mechanics, of the "start with Schrödinger's equation" variety, which means it's meaty math that's hard to do without powerful computers, and even then, you either have to make big, (over-)simplifying assumptions or deal with small fry. The systems where singlet fission takes place involve lots of conjugated bonds--a giant line of benzene rings smushed together, or maybe just a line of double bonds (the latter is what our eyes use to see light, FWIW). This provides a simplifying assumption for the math.
Now we come to what this paper is doing. This paper is swapping out one of the subroutines for the math with a quantum computer calculation. It's using a test molecule, and comparing the results of the quantum-based simulation with the purely-classical-based simulation. Note that everything here is pure simulation: there's no real, physical molecules being studied!
Because quantum computers that exist today are weak, they are using the simplest possible system for their work--this is the H₄ system. This H₄ is not a model of any real molecule [1]. Rather, it's a reduction of the behavior of real, interesting systems--linear conjugations of orbits--into the simplest possible model, in order to allow some of the behavior to even be studied in the first place.
So, in short, this is a paper that is concluding that a more powerful quantum computer might be helpful in doing the calculation work needed to evaluate candidate molecules that might make better solar cells. They've done this by showing that a quantum computer can indeed do the calculation on a simple model and that the results track existing classical computations (note there's no actual evaluation of if the quantum computer did it faster).
[1] Offhand, I'd say it's not what a real H₄ would look like. H₄ would likely be a tetrahedral complex. But I also imagine it's thermodynamically unstable and would dissassociate into multiple molecules with any number of electrons: neutral charge would definitely go to 2H₂, +1 charge to H₂ and H₂⁺, +2 charge probably to 2 H₂⁺. I've got no idea how the hell H₄³⁺ would break apart, but I have to imagine that one electron can't keep the protons from flying out of the molecule. In no case would you end up with a linear arrangement of atoms 2Å however
Best comment award!
The point of doing these toy simulations, easily done on classical computers, is to understand the quantum circuits that calculate molecular behavior so that when those circuits are generated for nontrivial molecules the results can be received with confidence.
Photon goes bonk on a molecule and it knocks an electron free (we can turn this electron into power in solar cells!). Sometimes, however, this electron knocks a second electron free, which is the "singlet fission" process being described (this means we get more power in our solar cells). We want to model this process in various molecules to be able to make better molecules for solar cells.
There's a problem... this is quantum mechanics, of the "start with Schrödinger's equation" variety, which means it's meaty math that's hard to do without powerful computers, and even then, you either have to make big, (over-)simplifying assumptions or deal with small fry. The systems where singlet fission takes place involve lots of conjugated bonds--a giant line of benzene rings smushed together, or maybe just a line of double bonds (the latter is what our eyes use to see light, FWIW). This provides a simplifying assumption for the math.
Now we come to what this paper is doing. This paper is swapping out one of the subroutines for the math with a quantum computer calculation. It's using a test molecule, and comparing the results of the quantum-based simulation with the purely-classical-based simulation. Note that everything here is pure simulation: there's no real, physical molecules being studied!
Because quantum computers that exist today are weak, they are using the simplest possible system for their work--this is the H₄ system. This H₄ is not a model of any real molecule [1]. Rather, it's a reduction of the behavior of real, interesting systems--linear conjugations of orbits--into the simplest possible model, in order to allow some of the behavior to even be studied in the first place.
So, in short, this is a paper that is concluding that a more powerful quantum computer might be helpful in doing the calculation work needed to evaluate candidate molecules that might make better solar cells. They've done this by showing that a quantum computer can indeed do the calculation on a simple model and that the results track existing classical computations (note there's no actual evaluation of if the quantum computer did it faster).
[1] Offhand, I'd say it's not what a real H₄ would look like. H₄ would likely be a tetrahedral complex. But I also imagine it's thermodynamically unstable and would dissassociate into multiple molecules with any number of electrons: neutral charge would definitely go to 2H₂, +1 charge to H₂ and H₂⁺, +2 charge probably to 2 H₂⁺. I've got no idea how the hell H₄³⁺ would break apart, but I have to imagine that one electron can't keep the protons from flying out of the molecule. In no case would you end up with a linear arrangement of atoms 2Å however