For my 20,000th post, I figured I ought to post this old thing from 2015.
What's the smallest chip that can do exaflop calculation?
When people say "exaflop computer" what's usually meant is "a computer that can do at least one exaflop per second."
This site says that, currently, the Chinese supercomputer Tianhe-2 is the fastest computer in the world, at 33.86 petaflops/second. What I'm going to do is figure out how many transistors are in it, find a transistor-to-flop ratio, and calculate the size of a theoretical exaflop chip based on that.
This Forbes article says that Tianhe-2 consistes of 16,000 nodes, "which each contain two Xeon IvyBridge processors and three Xeon Phi processors." That's good information, because we can find out how many transistors are in Ivy Bridge and Phi processors. This makes for a total of:
2 Ivy Bridge/node * 16,000 nodes = 32,000 Ivy Bridge processors
3 Phi/node * 16,000 nodes = 48,000 Phi processors
We don't have information on exactly what processor models Tianhe-2 is using (which I would guess is probably a Chinese state secret), but I'll assume that the most powerful/largest/highest transistor count models are used.
For Ivy Bridge, the highest transistor count model is the Ivy Bridge-EX, clocking in at 4.31 * 109 (4.31 billion) transistors.
For Phi, the highest transistor count model is the Xeon Phi SE10X, with 5 * 109 (5 billion) transistors.
So the Tianhe-2 has:
32,000 Ivy Bridge processors * 4.31 * 109 transistors/Ivy Bridge processor + 48,000 Phi Processors * 5 * 109transistors/Phi processor = 3.7791 * 1014 transistors
These 3.7791 * 1014 transistors are able to produce 33.86 petaflops/second, which is equivalent to 3.386 * 1016flops/second.
To find the ratio of transistors to flops:
(3.386 * 1016) / (3.7791 * 1014) = 3,386,000 / 37,791 flops/second/transistor
That's not a very round number, but it's roughly equal to 89.5981 flops/second/transistor.
Our goal is a computer that can do 1 exaflop/second, equivalent to 1 * 1018 flops/second.
Using our current flops/second/transistor ratio, this would take:
(1 * 1018 flops/second) / 89.5981 flops/second/transistor = 1.1161 * 1016 transistors
This is roughly the point in my calculations that the unit "flops" starts to sound pretty silly.
So we'll need a total of 1.1161 * 1016 transistors. That's 1.1161 quadrillion. That's a lot.
In your post you specified that each transistor should be three atoms large. I don't think that's super feasible with current technology (barring some hardcore cooling to keep everything working properly) but hey - nobody has made an exaflop computer yet either so we're talking hypotheticals anyway.
Ninja edit: Apparently three-atom transistors are totally a thing. The paper is behind a paywall so I can't access it but still; cool.
Another issue is that transistors need space between them in order to work properly. I can't really find how much space, but we'll assume that since our transistor technology has advanced to the reliable three-atom range, we need two atoms of space between each transistor. We'll also assume that each transistor is 3 atoms long by 1 atom wide. This means that each transistor takes up an area of:
(3 atoms + 2 atoms) * 3 atoms = 15 atoms2
This brings us to, finally, the size of our chip:
15 atoms2 /transistor * 1.1161 * 1016 transistors = 1.67415 * 1017 atoms2
As far as I'm aware, transistors are still made of Silicon and our hypothetical 3-atom transistor should be no different.
Silicon's atomic radius is 110 picometers, or 1.1 * 10-10 meters. This datatable says that a Silicon-Carbon bond is 186 picometers (or 1.86 * 10-10 meters) long. We'll assume that a silicon-silicon bond is similar enough that the same numbers apply.
So in one direction (call it the 'x' axis) we have five atoms and four bonds, and in the other direction (call it the 'y' axis) we have three atoms and two bonds. Our transistor dimensions are then:
5 atoms * 1.1 * 10-10 meters/atom + 4 bonds * 1.86 * 10-10 meters/bond = 1.294 * 10-9 meters in the x direction
3 atoms * 1.1 * 10-10 meters/atom + 2 bonds * 1.86 * 10-10 meters/bond = 7.02 * 10-10 meters in the y direction
You can pretty much think of this as a rectangle, and the area of that rectangle is:
1.294 * 10-9 meters * 7.02 * 10-10 meters = 9.08388 * 10-19 meters2
So 9.08388 * 10-19 meters2 is the size of each transistor.
Finally, the size of our chip:
1.1161 * 1016 transistors * 9.08388 * 10-19 meters2 /transistor= 101.4 cm2
That's roughly the size of a CD.
So, now, why is that number so small? Honestly it surprised me too, but the math is correct. The key thing here is that you specified that each transistor should be three atoms big. I added onto that to try and simulate how transistor might actually be placed on a chip, but still. Three atoms is really really really small.
Obviously we're not at the point yet that we can reliably produce huge quantities of three-atom transistors (at least not in the scale this exaflop computer would need).
But this question, and subsequent answer, shows exactly why tech sites get so excited about news of smaller transistors - once we can produce transistors that are so ludicrously small we're open to such huge advances in processing size that it's almost unbelievable.
tl;dr: 101.4 cm2 , or roughly the size of a CD. See immediately above for explanation why it's so small.
ZorbaTHut figured out:
Wolfram Alpha informs me that a cubic millimeter of silicon is 5*1019 atoms. This means you could fit a brain into a cube roughly 0.1mm on a side.
This would technically be considered the right size for sand, although it's real close to being considered silt instead. Call it very fine sand.
If it's graphene instead of silicon, that's a power fold increase of 102 or 103 (or more). Let's be conservative. So at 1/100th the size, the chip now only has an area of about 96mm2. Round that up a bit (it'll need plenty of heatsinks I'm sure) and it's 10mm to a side. About the size of a coffee bean. An exaFLOPS coffee bean!
However, this is still not an exaFLOPS nanobot you seem to be hoping for. Fear not, the journey is not over. Firstly, I don't think every single nanobot in your blood would have to be exaFLOPS rated. A whole fleet of quadrillions of nanobots could form a distributed system.
Also, the computing rabbit hole goes deeper than graphene! The nanobots could have molecular locks, or it could be 3D graphene, or atomic computation, or even subatomic computation! This would be so much theoretical computational power, your blood wouldn't even know what to do with it and the nanobots might get bored.
Nanomachines really don't need any more than megaFLOPS to be useful.
As for when we'll see such nanocomputers, well we're getting there.
Recall this fun thing:
This computer is smaller than a grain of salt, stronger than a computer from the early '90s, and costs less than 10¢. 128 of them together is still much smaller than the tip of your finger.
Also recall, from 2014:
Engineers Build World's Smallest, Fastest Nanomotor
One nanometer thick graphene mimics two stroke combustion engines, but without the side effects