Heat in a Quantum Computer

With regard to heat: everybody knows that computers generate a lot of heat. When you make computers smaller, all the heat that's generated is packed into a small space, and you have all kinds of cooling problems. That is due to bad design. Bennett first demonstrated that you can do reversible computingthat is, if you use reversible gates, the amount of energy needed to operate the gates is essentially indefinitely small if you wait long enough, and allow the electrons to go slowly through the computer. If you weren't in such a hurry, and if you used ideal reversible gateslike Carnot's reversible cycle (I know everything has a little friction, but this is idealized)then the amount of heat is zero! That is, essentially zero, in the limitit only depends on the losses due to imperfections.

Furthermore, if you have ordinary reversible gates, and you try to drag the thing through as quickly as you can, then the amount of energy lost at each fundamental operation is one kT of energy per gate, or per decision, at most! If you went slower, and gave yourself more time, the loss would be proportionately lower.

And how much kT do we use per decision now? 10¹⁰ kT! So we can gain a factor of 10¹⁰ without a tremendous loss of speed, I think. The problem is, of course, that it depends on the size that you're going to make the computer.

If computers were made smaller, we could make them very much more efficient. It hadn't been realized previous to Bennett's work that there was, essentially, no heat requirement to operate a computer if you weren't in such a hurry. I have also analyzed this model, and get the same results as Bennett with a slight modification, or improvement.

If this device is made perfectly, then the computer could work ballistically. That is, you could have this chain of electron sites and start the electrons off with a momentum, and they simply coast through and come out the other end. The thing is donewhshshshsht! You're finished, just like shooting an electron through a perfect wire.

If you have a certain energy available to the electron, it has a certain speedthere's a relation between the energy and the speed. If I call this energy that the electron has kT, although it isn't necessarily a thermal energy, then there's a velocity that goes with v, which is the maximum speed at which the electron goes through the machine. And when you do it that way, there are no losses. This is the ideal case; the electron just coasts through. At the other end, you take the electron that had a lot of energy, you take that energy out, you store it, and get it ready for shooting in the next electron. No losses! There are no kT losses in an idealized computernone at all.

In practice, of course, you would not have a perfect machine, just as a Carnot cycle doesn't work exactly. You have to have some friction. So let's put in some friction.

Suppose that I have irregularities in the coupling here and therethat the machine isn't perfect. We know what happens, because we study that in the theory of metals. Due to the irregularities in the positions or couplings, the electrons do what we call "scattering." They head to the right, if I started them to the right, but they bounce and come back. And they may hit another irregularity and bounce the other way. They don't go straight through. They rattle around due to scattering, and you might guess that they'll never get through. But if you put a little electric field pulling the electrons, then although they bounce, they try again, try again, and make their way through. And all you have is, effectively, a resistance. It's as if my wire had a resistance, instead of being a perfect conductor.

One way to characterize this situation is to say that there's a certain chance of scatteringa certain chance to be sent back at each irregularity. Maybe one chance in a hundred, say. That means if I did a computation at each site, I'd have to pass a hundred sites before I got one average scattering. So you're sending electrons through with a velocity v that corresponds to this energy kT. You can write the loss per scattering in terms of free energy if you want, but the entropy loss per scattering is really the irreversible loss, and note that it's the loss per scattering, not per calculation step [heavily emphasized, by writing the words on the blackboard]. The better you make the computer, the more steps you're going to get per scattering, and, in effect, the less loss per calculation step.

The entropy loss per scattering is one of those famous log₂, numberslet me guess it is Boltzmann's constant, k, or some such unit, for each scattering if you drive the electron as quickly as you can for the energy that you've got.

If you take your time, though, and drive the electron through with an average speed, which I call the drift speed, v_D (compared to the thermal speed at which it would ordinarily be jostling back and forth), then you get a decrease in the amount of entropy you need. If you go slow enough, when there's scattering, the electron has a certain energy and it goes forward-backward-forward-bounce-bounce and comes to some energy based on the temperature. The electron then has a certain velocitythermal velocityfor going back and forth. It's not the velocity at which the electron is getting through the machine, because it's wasting its time going back and forth. But it turns out that the amount of entropy you lose every time you have 100% scattering is simply a fraction of kthe ratio of the velocity that you actually make the electron drift compared to how fast you could make it drift. [Feynman writes on the board the formula: k(v_D/v_t).]

If you drag the electron, the moment you start dragging it you get losses from the resistanceyou make a current. In energy terms, you lose only a kT of energy for each scattering, not for each calculation, and you can make the loss smaller proportionally as you're willing to wait longer than the ideal maximum speed. Therefore, with good design in future components, heat is not going to be a real problem. The key is that those computers ultimately have to be designedor should be designedwith reversible gates.

We have a long way to go in that directiona factor of 10¹⁰. And so, I'm just suggesting to you that you start chipping away at the exponent.

Thank you very much.