cohomologo

Note this is version 2 of the Arxiv posting. The first version claims a proof of Collatz, which appears to be effectively withdrawn by the update.

Physicist here:

Just gave it a cursory look, but looks like crank nonsense to me. The paper makes no effort to give any context on what an informaton at a level that physicists would understand.

True, but as someone who works on quantum physics using classical computers and loves to see classical computers simulating increasingly hard quantum problems, these "crushings" were still only done on classical computers in a way that would scale exponentially with the number of qubits.

Even then, I believe the classical simulations to beat google's 53 qubit device were never actually performed -- it was shown that they could be performed quickly with petascale memory but actually doing it would be a huge expense. Add a dozen more qubits and even the hypothetical classical challengers fall off quickly...

I research QC. Feel free to explain how my skills could be used to meaningfully contribute to the problem of world hunger or renewable energy while also keeping me employed --- going back to college to train in a different field isn't feasible. If it's not so easy, then perhaps it's logical that QC researchers and AI researchers and software devs aren't all jumping ship to work on "more important topics"...

You may just be spouting off, but I genuinely am asking. If someone else dissing QC research wants to make a pitch for a concrete plan on how to make a difference in the world with a physics PhD and years of experience in scientific computing, drop me a message.

No, but the quantum groups are making progress towards essentially making 1 logical qubit -- a common scheme uses a square grid of physical qubits, say with size 7x7, 8x8, 9x9, etc.. as a single logical qubit. This may seem like a moving target, but it is not. There is a specific, concrete error rate on the physical qubits, below which it is known that the logical qubits on increasingly large grids get _exponentially_ more accurate. Once the engineering achieves that noise level, logical qubits in the form of NxN grids where N~10-20 will ~eventually~ follow. I won't venture to predict how long, as building and controlling these grids of qubits while keeping the error rate low isn't easy. Could be years or decades.

In my experience in QC, I've yet to see many grifters. In fact, many of the people in quantum computing are quite open about the fact that we aren't that close to "useful" quantum computing, that the number of practical problems that quantum computers will beat classical computers with could be quite small, and are actively worried about over-hyping. People selling today's QCs as a solver for generic optimization problems are indeed grifting, but that is not most of the community.

Despite all these problems, myself and much of the community still think QC is worth attempting --- for my part, the applications to quantum physics is the main motivation, and one in which it's relatively certain that QC will not "become obsolete". (It's also still perfectly valid to _research_ how to make QC useful in various types of classical problems, including optimization, and it's plausible that progress _could_ be made that would open up more widespread uses of QC. The line, for me, is when people misrepresent the likelihood of success of that research.)

As far as I understand it, either two incommensurate frequencies or the Fibonacci sequence ABAAB… approach produce similar physics. The Fibonacci sequence is easier to simulate numerically on a (classical) computer because there is a recursive property to it that allows you to jump forward in time in large steps, making it nice for theorists even if the experiments are fairly similar.

I got lucky with 5 as well:

ADIEU BLAST MANOR FANNY CANNA

⬜⬜⬜⬜ ⬜⬜⬜⬜ ⬜⬜⬜ ⬜⬜

With double exponential growth, “it looks like nothing is happening, nothing is happening, and then whoops, suddenly you’re in a different world,” Neven said.

This is just a clever way to spin the fact that we are experiencing growth much slower than exponential growth now into a prediction of much faster future growth, without any evidence. Or perhaps an internal joke the physicists would make. Next time I have a really flat function, I'm going to fit it with a triple exponential like so:

  https://www.wolframalpha.com/input/?i=plot+exp(exp(exp(x))),+x%3D-20..-5,+y%3D0..5

Is "beyond a shadow of doubt" the standard of proof we need to take action? I would advocate for an actuarial standard. (i.e. estimate the probabilities and costs/benefits of various outcomes and take the path with the best expected outcome, while continually updating your probability estimates with new information.) The issue with climate change is not that it is 100% certain, but that the responses require globally coordinated action where every nation is incentive to cheat; and as with other types of economic transition, there are huge inefficiencies if we try to transition too fast (i.e. global recessions required to build the necessary infrastructure as in https://dothemath.ucsd.edu/2011/10/the-energy-trap/). So the downside risk if we fail to act and climate change is real is tremendous.

From the ARCH foundation FAQ:

Beyond the laser based digital layer there may also be layers that require the ability to detect and decode molecular, atomic scale or subatomic scale (quantum or holographic) information. We can already encode data in this way, but reading it requires very advanced technology.

They seem so clueless that even if I bought into the underlying mission of preserving data by sending it into space, I wouldn't trust this group to do it.

I think you've got that last part backwards. When you have constituents with binding energy, that reduces the total energy of the system and reduces the mass. So it seems that the mass of the proton is quark mass + gluon kinetic energy - bonding energy, and most of the mass comes from gluon kinetic energy.

It's pretty easy. If your computer has enough ram to store a size 2^16 length complex vector (which is 2^20 bytes, or 1MB) than you can open up an ipython notebook and write code to apply quantum gates to it with no problem.

The problems start to set in if your RAM can't hold the wavefunction in memory (so around 28 qubits, which takes 2^32 bytes = 4GB of RAM.)

With specialized code and supercomputers you can get a little farther, but you will be fighting exponential growth, so not too much. The practical limit for classical computers is in the 40-50 qubit range.

I think it's more like a 6 bit number, and probably not very fast at all...

The Google group in Santa Barbara is building to a 49-qubit computer. If I understand correctly, it'll be interesting for understanding how you calibrate your quantum machine and validate that it is doing what it is supposed to. The algorithms they run on classical computers to validate that device will need a decent amount of supercomputer time. And much bigger devices would be impossible to validate with a classical computer in the same way, so in some sense you hit "the limit" of classical computation and start to move beyond.

But for the computational problems that are useful for applications, which are not very much like the problem they use for validation, 49 qubits is still far far away from beating a classical computer.

Not any given quantum computer -- Scott Aaronson debunks the D-Wave "quantum" annealers that aren't actually shown to be harnessing the quantum power.

IBM's machine by contrast is a genuine quantum computer -- the kind Scott Aaronson would probably have no problem with. But the number of qubits is too tiny to do anything interesting.

The point where it gets interesting for realistic physics and chemistry applications is around 100 (error-corrected) logical qubits and 10^8 coherent operations, see for example https://arxiv.org/abs/1510.03859.

The error correction adds another factor of at least 100 or so in both qubits and gates needed (but possibly much bigger than 100, depending on qubit quality), see for example https://arxiv.org/abs/1312.2316.

Other fields of application - factoring large integers, for example - takes many many more qubits to be interesting.

While it's good to get people excited about the potential of quantum computing, it's seems a bit disingenious to suggest that a 17-bit quantum processor is commercially interesting. I especially like how they juxtapose it with the publically available 16-bit quantum processor to make it seem like one extra qubit makes it worth paying money...

> As it does, its power to make predictions of unknown objects should grow, until we've reached the limit of our current instruments' ability to observe.

Well that's kind of the problem already. For the objects that are interesting, we've either discovered them or they are hard (or impossible) to discover with current instruments. For the ones that are merely hard but not impossible, the process of finding them requires human cleverness or significant telescope resources (and human cleverness can be applied to figure out the best way to direct the limited resources). I'm not really sure that automation helps...

The symmetry breaking should be thought of as a phase transition that occurs as the temperature of the universe changes, like liquid freezing and becoming ice. The universe was initially very hot, but rapidly cooled down as it expanded and went through phase transitions when it passed the "freezing temperature", i.e the temperature at which the laws of physics prefer to spontaneously break the symmetry.

In fact, treating particles like vortices and vice versa is a good analogy that can be made more precise with an idea called particle-vortex duality. The basic idea is that you make a change of variables so that the vortices of the old variables become the particles of the new variables and vice-versa.

Nope. Like in the ground state of the hydrogen atom, the ground state of the quantum harmonic oscillator has the wave function spread out in a sphere, and this wave function stays constant in time. Almost all quantum systems do something like this in the ground state, which shows why the time crystal is something new and interesting.

Electrons do not orbit atomic nuclei in the quantum description of atoms. Instead, in the lowest energy state the electron wave function is spread out in a sphere around the nucleus, and this wave function stays constant in time. Almost all quantum systems do something like this in the ground state, which shows why the time crystal is something new and interesting.

The connection with quantum information (the theme of research at Station Q) is through the many-body localization (MBL) aspect of this system. These are phases that store quantum information at finite temperature. Normal quantum systems thermalize and lose any information of their initial state. (Actually, the information spreads chaotically throughout the system and becomes hard to retrieve, so its effectively lost.) In MBL systems, the information does not spread, and the system does not thermalize. You can read the information of the initial state back out after waiting a long time.

Physicist here with some context:

The systems considered here have periodic drives (in the article, "Floquet"). This means that time-translational symmetry is already partially broken. The system is only the same after waiting times that are multiples of the period T of the drive.

The time-translational symmetry breaking occurs because the state of the system is not periodic with period T as would normally happen but periodic with period 2T.

In terms of frequencies, if the drive frequency is f = 1/T, then this system responds at a frequency f/2, whereas normal systems can only respond at frequencies f, 2f, 3f, ... that correspond to harmonics.

Additionally, this time-translational symmetry breaking makes a stable phase of matter -- that is, you don't have to fine tune any parameters of the system to see the effect, and experimental noise won't destroy it. It also doesn't matter which initial state you prepare your experiment in. While not as exotic as a time-translational symmetry breaking without a drive to partially break the symmetry first, it is still pretty surprising that this type of phase exists at all. It is likely that spontaneous breaking of full time-translational symmetry can never be stable in the same sense.

Knot theory is still quite popular with physicists of certain sorts, such as condensed matter physicists that study "topological phases" - a popular account that I like can be found here: https://www-thphys.physics.ox.ac.uk/people/SteveSimon/PWsept...

Cool, thanks for the info. I enjoyed playing with Quirk, it's very nice.

I've never seen a physics book that treats this using the zeta function, except popular articles that try to present this calculation as mysterious. In practice, in my QFT class we needed to compute the sum

lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),

that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form

sum = 1/epsilon - 1/12 + O(epsilon).

The 1/epsilon term (which goes to infinity) drops out of the final physical result when you do the calculation properly.

There is a much simpler mathematical treatment of this series that appeared in my class on quantum field theory in the computation of the vacuum energy. In that case, the series was treated as

lim \epsilon -> 0+ ( \sum_{n=1}^\infty n e^{- \epsilon n} + ...),

that is the series was multiplied by a decaying exponential function with a rate of decay that goes to zero. This sum can easily be evaluated for small epsilon takes the form

sum = 1/epsilon - 1/12 + ...

Crucially, there was another term in the calculation that naturally appeared that canceled the 1/epsilon. Without that other term, the sum would of course be infinite when epsilon -> 0.

This is much simpler than analytic continuation through the complex plane, and again, this is how the physics calculation appears in QFT courses. There is no need to appeal to complex analysis here, which leads to all of this mysticism and confusion.

Could you help us understand what the limiting factor in these simulations are?

By memory considerations alone, a N-qubit wavefunction (using 64-bit floats) uses 2^(N+4) bytes, and a N-qubit unitary operator uses 2^(2N+4) bytes. If you use 1 GB of RAM, that allows you to store full unitary operators up to 13 qubits.

If you use sparse operators to store the gates (which have a size in memory that is a constant times the wavefunction size) you can imagine doing 24 qubits.

Of course fighting exponential scaling is always hard, but I'm not sure if I understand why the limit (for a hobby-level project) is closer to 10 than 24.

Some of the operations that are performed on the tensors in a neural network are non-linear. An example might be taking the tanh of all of the elements of the tensor. For these steps, you won't have invariance (or covariance) under change of basis.

Even in physics, there are applications of tensors which essentially treat tensors as multidimensional arrays (see for example, tensor networks) with no predefined transformation properties. But the operations done on tensors are always linear.