Why classical simulation explodes
Simulating a quantum system on a classical computer requires tracking a complex number for every possible state the system could be in. For n two-state particles, that means 2-to-the-n numbers. Fifty entangled particles demands around 10-to-the-15 numbers; 300 particles demands 10-to-the-90, which exceeds the estimated number of atoms in the observable universe. This is why brute-force classical simulation of even modestly sized quantum systems, like a protein's electrons, is not just slow but physically impossible.
- Krishna offers a handy conversion: 2-to-the-10 is roughly 10-to-the-3, so powers of 2 can be quickly rewritten in base 10 by multiplying the exponent by about 0.3.
- The state vector cannot simply list each particle separately because quantum particles are indistinguishable, so you must track linear combinations of all possible arrangements, not individual labeled states.
- Krishna notes, as an aside, that if every one of the roughly 10-to-the-80 particles in the universe were a two-state system, the universe itself would need to track 2-to-the-(10-to-the-80) complex amplitudes, a number that makes the simulation problem feel philosophically unsettling.
Transcript
This chapter, from the episode video's captions · 880 words
43:41like vector tensor or like set of numbers across all 2 to the n computational states for Alice and Bob for example, right? You could have up up. You could have down down. You could have up down or down up. And those are different >> across both the Z and the X. >> Well, here I'm just I'm just saying like >> even if you keep it even if you just keep it that simple. >> Yeah. Even if we just keep it that simple right? >> Okay. Right. Which like the point being it's not >> Yeah. Anyway. >> Yeah. The Z and the X comes in like a bit later and I'll have to get you know because like they're all like like Yeah. Whatever. In any case, like let's let's just say two two particle states, right?
44:22Two state systems, right? Those two states can actually go in Z and X is is is the point that I was trying to make. But but um >> what you need to do is for Alice and Bob, you need to keep track of four possibilities. The up, the down, the down up, and the and the and the up down. Really, you got to take take um take care of like the sum of this and the difference of this because like you know what does it mean to be like the the electron one is up and electron 2 is down? There's no sticker on an electron saying this is one and two. They're indistinguishable particles. So you have to worry about like you know combining like linear combinations of them. But in any case it's always 2 to the n >> is the idea. Now this becomes
45:03insurmountable very very quickly. very very quickly. Okay, if you want 50 entangled two-state particles, that's 2 to the 50 complex numbers that you have to take care of, right? 2 to the 50 um hack for students that are listening. Um 2 10 is like 10 3 because 2 to the 10 is 10 24 which is 10 3. So if you ever want to go into base 10, 2 is like 10 3 to the 5 which is 10 15. So you got to take care of 10 15 complex numbers. If you're trying to sim simulate 300 two-state particles, which like any there's so many compounds where there's 300 electrons that are moving around, proteins for example, um that's going to
45:43be 2 300, which is uh 2 10 to the 30, which is 10 3, which is 10 90 numbers. That exceeds the number of elementary particles in our observable universe. There's estimates out there that there are only 10 the 80 atoms, right? So just to simulate 300 two-state particles, I need 10^ the 90 numbers. And it's kind of crazy. The part of the point you're bringing up here is that it is just wholly inefficient >> because the type of systems we would be able to simulate are exceedingly small, effectively a protein. Yeah.
46:23>> That only had two states, which is not real. Like it's not practical in real life. >> Yeah. No, I'm saying a protein would be would be crazy. I mean, maybe you could simulate if you had all the time in the universe like a water molecule with 10 with with 10 electrons, right? there there's no amount brute force um around compute and scaling up that would make it even tenable because >> the amount of variables when we have to account for the entire system as a whole uh because you can't factor >> factor it means that the amount of uh variables and interactions you're
47:03tracking continuously is just is just insane. Yeah. And and also as an aside, I think it's kind of cool to think about that like somehow the universe is keeping track >> of what? So there's 10 80 particles, right? Even if all of those particles are two-state systems, there's two there's 2 to the 10 the 80 complex amplitudes that the universe is keeping track. I don't even know how big that number is, right? That might be I don't know if that's the big No, I'm sure there's mathematicians that have come up with bigger numbers than two to the 10 to the 80. Yeah. >> But I'm just saying like it's kind of crazy that the universe is keeping track of all of those complex amplitudes to
47:44give us like the world. Is that how it works? Really? Is that really how it works? Sometimes I'm thinking about this stuff and I'm like, mate, is this really how it works? >> It's it's it's cuz it's hard to even >> and like where anyway. Yeah, but that's that's an aside about like just like the nature of reality and like totally totally totally >> why quantum mechanics is weird. But in any case, right, this is what Fineman is talking about and he says instead of classical bits where we'd have to we'd have to create 10 the 90 bits to keep track of 300 thingies. Um instead, what if we use a quantum version of a bit, right? What if we create a computer
From How Quantum Computing Actually Works (Part 1)
Part I of our quantum computing deep dive traces the field from Bell and Feynman to Deutsch and Shor—and explains what quantum computers actually do differently from classical machines.