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A new quantum algorithm for classical mechanics with an exponential speedup

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

#51

Post by zopa »

smallnamespace wrote: post #46
ndriscoll wrote: post #45

Earlier quoted context omitted.

Look around you. Is the building you're in falling down? Or is it standing still? Are the things on your desk bouncing about, or are they just sitting there? Are the atoms in your body fissioning, or are you still here? Stable systems are extremely common. The world would be a much more violent place to live in if they weren't. And it makes sense that stable points exist. That means there's a well somewhere in the po…

I think your explanation only seems clear because you’re eliding some quite relevant detail > Look around you. Is the building you're in falling down? You can say it’s stable, or that the building is in the act of slowly falling down without human input (maintenance). It’s a bit misleading to sneak in time scale. > Are the things on your desk bouncing about, or are they just sitting there? At a small enough scale bit…

This is where we need to decide if we're doing physics or philosophy. Physics is about answering specific, fairly practical questions, and the questions tell you what parts of reality you can afford to ignore. If you want to get at the true nature of things in a metaphysically satisfying sense, physics will always disappoint you.

Is it part of the essential nature of a building to collapse? Sure, I suppose, as with all things. But I'm only going to be in this this coffee shop for another hour: to me, for my purposes, it's stable. If instead I were buying the building I'd want a much more detailed model that considered termites and the risk of earthquakes and all sorts of things, but I still wouldn't care about the date of the next ice age that will scour the landscape clean.

One thing to keep in mind is that our mathematical methods might as well be approximate, because we our measurements always will be. Precise calculations on fuzzy data are a waste of time. Not that there's anything wrong with philosophy! But it's not super relevant to understanding where harmonic oscillators are a useful model.

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

#52

Post by somat »

IIAOPSW wrote: post #21

I'm skeptical. If you have an exponential speedup for simulations of coupled oscillators, you can rig a system of coupled oscillators into a general purpose computer [0] and therefore have an exponential speedup for any computation. That seems too good to be true. [0] https://www.zyvex.com/nanotech/mechano.html

I am probably misunderstanding something fundamental. But isn't that exactly what a qubit is. A coupled harmonic oscillator.

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

#54

Post by abdullahkhalids »

pa7x1 wrote: post #17
abdullahkhalids wrote: post #4

One of the most important insights you take away from a physics undergrad is that you can model much of physical phenomena as a harmonic oscillator. The reason for this is quite simple 1. Every closed system has a fixed total energy, so many systems just settle into an oscillating state, where kinetic energy converts into potential and back. 2. Most real world systems are approximately closed, so they leak energy til…

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. Zeroth-order term (constant) does not determine the dynamics. The first-order term (linear) is zero only for low energies, as any potential well at sufficiently low energies will be symmetric. The second-order term (quadratic) is what provides the restoring force towards the equilibrium.

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

#55

Post by mjburgess »

ndriscoll wrote: post #47
smallnamespace wrote: post #46

Earlier quoted context omitted.

I think your explanation only seems clear because you’re eliding some quite relevant detail > Look around you. Is the building you're in falling down? You can say it’s stable, or that the building is in the act of slowly falling down without human input (maintenance). It’s a bit misleading to sneak in time scale. > Are the things on your desk bouncing about, or are they just sitting there? At a small enough scale bit…

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 far as research goes. Describing beds is not a pressing theoretical quesiton

Describing organic systems, say is -- chaotic organic development across trillions of cells. There are no SHOs there

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

#56

Post by godelski »

pa7x1 wrote: post #17
abdullahkhalids wrote: post #4

One of the most important insights you take away from a physics undergrad is that you can model much of physical phenomena as a harmonic oscillator. The reason for this is quite simple 1. Every closed system has a fixed total energy, so many systems just settle into an oscillating state, where kinetic energy converts into potential and back. 2. Most real world systems are approximately closed, so they leak energy til…

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.

I don't think fewer words means less complicated. I know we're on HN but I don't think most of the crowd here is going to be familiar with dynamic systems to parse your words and certainly not DEs or PDEs. I mean you won't get to this till what, Classical Mech in 3rd year of a physics undergrad? Maybe I'm too pessimistic but I feel like having sufficient background to parse your statement implies sufficient background to already have this knowledge. I definitely think OP's version is far clearer to the layman.

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

#57

Post by godelski »

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…

Honestly I kinda laughed when I saw the response because it reminded me of the classic nerd flexing (that happens especially when conveying something to laymen) where two nerds go at it saying the same thing but using heavier and heavier jargon as a means to flex (increasingly losing the laymen) rather than clarify or admit that language requires inference and the whole show is ironically just a demonstration at accurate communication or else one would not have been able to "um acktually" the other. (I think a lot of us have been guilty of this, often unintentionally, but we can still laugh at ourselves)

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

#58

Post by godelski »

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?

Most things jiggle even if you don't poke them. In fact, for them to not jiggle you'd have to be at absolute zero (not technically correct, but good enough. Damn you quantum and your jigglyness)

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

#59

Post by godelski »

mjburgess wrote: post #44
ndriscoll wrote: post #43

Earlier quoted context omitted.

You can model any conservative system with stable equilibria this way (i.e. anything with local minima in the potential). As another poster pointed out, just do a Taylor expansion around the equilibrium state, and you have a locally quadratic potential, i.e. a harmonic oscillator. Explicitly, WLOG the potential is `V(x) = V_0 + V_1 (x-x_0) + V_2 (x-x_0)^2 + ...`. But the first derivative is 0 near a minimum, so V_1 =…

How much of reality is a stable system under equilibrium?

The vast majority of it. By definition. Unstable equilibria are unstable to perturbations. By definition.

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

#60

Post by eigenket »

shollos wrote: post #50

I suspect an analog computer would work just as well for modeling coupled harmonic oscillators.

That would be very surprising. They show in this work that their problem is complete for the complexity class BQP. That means that if you can solve it on a classical computer (analog or not) in polynomial time you get (for free) classical polynomial-time algorithms for solving a bunch of problems we don't currently have classical poly-time algorithms for.

Most surpisingly this would include the hidden subgroup problem and hence give you a classical poly-time algorithm for integer factorization.

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