A new quantum algorithm for classical mechanics with an exponential speedup • Hacker News Board

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#61

Post by ridiculous_fish » Wed, Dec 06, 2023, 10:43 PM UTC

arbitrandomuser wrote: post #29

robertlagrant wrote: post #25

Earlier quoted context omitted.

> It's basically taking the first order perturbation of a Taylor expansion around a local minima. I'm not sure that's less complicated, at least for my level understanding.

> > It's basically taking the first order perturbation of a Taylor expansion around a local minima. > > I'm not sure that's less complicated, at least for my level understanding. You know Taylor expansions ? So every complicated function of x can be written as soe expansion of infinite powers of x, a0 +a1 ( x )+a2 ( x^2 ) ... And so on Every complicated force can then be written as a function of displacement x like t…

This is on the right track but not entirely right. In your F = a1x + a0, here a0 is a force, not a position, so you cannot neglect it on the basis of "default position." Instead you set it to zero because the system is at a stable equilibrium.

Here's the presentation I've seen. Usually we like to work in potentials, not forces, because potentials are nice scalar functions, while force is an ugly vector function. So say you have a potential V which is at a local minimum at position x.

Expand the potential at x around a small displacement dx: V(x + dx). This gives us the Taylor series V(x+dx) = V(x) + a1 V'(x) dx + a2 V''(x) dx^2 + a3 V'''(x) dx^3...

We can neglect V(x) since it's just a constant, and adding a constant to the potential does not affect the physics. And (the crux) we can neglect V'(x) because the potential is at a minimum, so the derivative is zero.

That leaves the quadratic and higher-order terms. Neglecting the higher order terms on the basis that dx is small, we get the harmonic potential, or Hooke's Law in the language of forces.

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#62

Post by taneq » Wed, Dec 06, 2023, 11:33 PM UTC

thfuran wrote: post #28

robertlagrant wrote: post #25

Earlier quoted context omitted.

> It's basically taking the first order perturbation of a Taylor expansion around a local minima. I'm not sure that's less complicated, at least for my level understanding.

Most things jiggle for a while if you poke them a bit?

Yep pretty much. Everything's jiggling a bit, if you poke them they jiggle, if you poke them twice has hard they jiggle twice as much, if you don't then they gradually slow down.

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#63

Post by noduerme » Thu, Dec 07, 2023, 1:07 AM UTC

fgoesbrrr wrote: post #20

noduerme wrote: post #9

Earlier quoted context omitted.

My lay understanding of the problem with classical algorithms is basically that a lack of resolution means you need to monte carlo the thing millions of times... which is why it's slow. If you could model it as a set of quantum states of similar inaccuracy, wouldn't that by definition be just as (in)accurate but faster? [edit] this reminds me of something I read about how NASA doesn't predict solar eclipses by trying…

People were predicting solar eclipses thousands of years ago with no calculators or even a modern understanding of math. Can't be that hard.

I found the article I was referencing[0]

>> In summary, it is clear ancient people could predict timings for lunar eclipses and partial solar eclipses, but there is no convincing evidence of people predicting the times and locations of total solar eclipses.

>> Today, we don’t rely on calculating the orbits of the whole Solar System to predict eclipses. For example, NASA uses a highly advanced form of an ancient technique – pattern recognition. Using some 38,000 repeating mathematical terms, NASA can predict both solar and lunar eclipses for 1,000 years into the future. Beyond that, the Moon’s wobble and Earth’s changing rotation make eclipse prediction less accurate.

[0] https://www.astronomy.com/observing/humans-have-been-predict...

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#64

Post by hughesjj » Thu, Dec 07, 2023, 1:45 AM UTC

mjburgess wrote: post #55

ndriscoll wrote: post #47

Earlier quoted context omitted.

Sure but the point is in a closed system, thermodynamic fluctuations alone are going to take eons to make your house fall down. What will actually get it is that the system is not fully closed, and eventually something from outside the system will give it the activation energy needed to escape the well it's in (e.g. a windstorm causing a tree to fall on it). But it's useful and practical to think of that larger syste…

Almost all of reality isnt day-to-day life --- indeed what that describes is, in many ways, exactly the sort of illusions of stability that admit cute mathematical analysis. You're weighting parts of reality by their relevance to a us at a particular place and time -- without such prejudice you find that very little admits of this sort of cute mathematical description. And that which does is now pretty exhausted as f…

Really? What about the Krebs cycle?

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#65

Post by mjburgess » Thu, Dec 07, 2023, 9:01 PM UTC

hughesjj wrote: post #64

mjburgess wrote: post #55

Earlier quoted context omitted.

Almost all of reality isnt day-to-day life --- indeed what that describes is, in many ways, exactly the sort of illusions of stability that admit cute mathematical analysis. You're weighting parts of reality by their relevance to a us at a particular place and time -- without such prejudice you find that very little admits of this sort of cute mathematical description. And that which does is now pretty exhausted as f…

Really? What about the Krebs cycle?

There is no first order taylor series expansion of the krebs cycle -- indeed not.

As soon as you have three of anythign physics, as applied math, breaks down

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#66

Post by pa7x1 » Sat, Dec 09, 2023, 2:29 PM UTC

lagrange77 wrote: post #38

pa7x1 wrote: post #17

Earlier quoted context omitted.

This is overly complicated. The reason harmonic oscillators pop up everywhere is even simpler and more general than that. It models first order perturbations over a stable equilibrium. For sufficiently small perturbations around a stable equilibrium everything is an harmonic oscillator. It's basically taking the first order perturbation of a Taylor expansion around a local minima.

> It models first order perturbations over a stable equilibrium. For sufficiently small perturbations around a stable equilibrium everything is an harmonic oscillator. That's the same reason, why we linearize nonlinear systems around the equilibria to apply linear control theory, right? While in control, this makes sense to me, since the goal is often to stabilize the system, how does this help with modeling the whol…

Yes, near a stable equilibrium you can always linearize a system to model its behavior quite accurately for small perturbations. That's the crux of what it was pointed above.

Outside of equilibrium things get more complicated. In principle, you can still do a first order expansion to understand the dynamics in a vicinity of that regime, the problem is that outside equilibrium you are not going to stay near the point of the solution space you started at. You will keep drifting, at least until you reach another stable equilibrium if there is one.

Systems outside equilibrium are much harder to study because we cannot linearize. Basically.

Re: A new quantum algorithm for classical mechanics with an exponential speedup

#67

Post by pa7x1 » Sat, Dec 09, 2023, 2:33 PM UTC

abdullahkhalids wrote: post #54

pa7x1 wrote: post #17

Earlier quoted context omitted.

This is overly complicated. The reason harmonic oscillators pop up everywhere is even simpler and more general than that. It models first order perturbations over a stable equilibrium. For sufficiently small perturbations around a stable equilibrium everything is an harmonic oscillator. It's basically taking the first order perturbation of a Taylor expansion around a local minima.

That's what I said, without using the word Taylor expansion, as this audience might not necessarily know it. I also explained in points 1 and 2 why many systems have can be modeled via small perturbations (i.e. Taylor approximation) around a stable equilibrium - they need to have low energy. Also, just a reminder that it is the second-order term in the Taylor expansion that is relevant for harmonic oscillators. Zerot…

The basic difference is pointing that there is something more fundamental and it doesn't have to do with physics. It has to do with math, with calculus, and with approximating complex functions via perturbative expansions. This is very general and applies to other kinds of systems, not necessarily physical. That's the key insight.

The first relevant term is quadratic in the potential around a stable equilibrium but linear in the force. That's why they are called linear harmonic oscillators, I thought that was obvious in my description.