This comment by Matt Kelly seems to be the most pertinent
> Regarding the similarity to a double slit interference pattern, what I found in my investigation (see my previous comment) is it was solely the result of the sorting algorithm used for the X coordinates of each possible outcome/weight. Ordering by the average X location results in a binomial distribution, but there are still ties to deal with. Applying secondary sorting to those based on the position of the left or rightmost X creates secondary binomial distributions within the main binomial distribution, similar to what Jonathan showed, except without all the spikiness because his example wasn’t consistently sorted in this manner.
> You can see the sorting I’m talking about by looking at the sorting of Xs in the output of weights in Jonathan’s double-slit examples. The bottom half is sorted as I stated, but the top half is somewhat out of order, corresponding to the spikes in the tallest curve in the center of the graphs.
> Maxime C, I had the same question as you – Why choose this sorting method for encoding the photon positions? My conclusion is that it doesn’t work however. Each additional step run with a large string produces an additional secondary binomial distribution. To get a smooth curve, we’d need to run billions of steps on a large string, but then we we’d also see billions of peaks, which no longer compares well with the expected diffraction pattern.
This is from four years ago, and there are some comments on the article by Matt Kelly that show that the "interference pattern" is just an artefact. The author did not respond despite several requests to do so.
I often wonder how much the fact that quantum mechanics' original formulation was in terms of a wave function and differential equations has to do with the ubiquity and importance of these topics at that particular era.
For example, Werner Heisenberg's doctoral thesis[1] arose from a contract of his doctor father Arnold Sommerfeld from a company that dealt with the channelling of the Isar river through the city of Munich. Very practical problems involving differential equations - kind of the bread and butter of physicists and engineers at the time.
What if quantum mechanics was found today in a world where the bread and butter has shifted to computer science, linear algebra and discrete math? Would we still end up with waves and differential equations, or would another formulation arise more naturally?
EDIT: I think a beautiful (but imperfect) example to illustrate this dichotomy in the ways of thinking is how the Bell inequality can be approached with photons and polarization or as a game. Thinking about Alice and Bob or polarized light, which would you prefer?
It was formulated at least twice in different versions. First by Heisenberg with his matrix mechanics and shortly afterwards by Schrödinger in terms of wave-functions.
There was considerable disagreement between the factions of physicists who favoured the different versions which essentially ended when after some considerable theoretical effort (mostly by Dirac) it was shown that the two pictures are exactly equivalent.
Physicists still use whichever formulation is most suitable for whatever problem they're trying to solve, for example if you're analysing the something where you care about a bunch of bound states like the simple harmonic oscillator or the hydrogen atom then the matrix picture tends to be easier to work with.
You are right that wave mechanics was more popular than matrix mechanics because physicists were already very familiar with wave methods.
Is it really true that we don't really understand why transformers work so well?
I mean we obviously understand how they work at a pure mechanical level, and we have this analogy with lookup (keys, queries, values) and "attention," but do we really get it? Can someone explain to me why that design works so much better than lots of other things like RNNs?
Or did we just tinker a lot (a method known as "graduate student descent") guided by mathematical hunches and loose analogies with biological brains until we found something that kinda worked?
It wouldn't be the first time. AFAIK we got the idea of wings from birds and figured out how to fly with them before we had a really solid fluid mechanical understanding of how and why wings work the way they do. We just thought "hmm so birds fly, so lets try stuff that looks a bit like that..."
I would like to stress what is also mentioned in a sibling comment: quantum mechanics' original formulation was not in terms of a wave function and differential equations.
Heisenberg's matrix mechanics was the first, and formulated entirely in terms of linear algebra. See for example the introduction of basic linear algebra techniques in the famous Bohr-Jordan paper from 1925:
For me it's a reminder that physics describes how quantum systems evolve, but it doesn't (in a sense) tell us what those systems are. Are particles waves? Matrices? Excitations of a field?
Each of these descriptions works, so I can't exclusively say any one of them is what particles really are. Bring a macroscopic being is philosophically frustrating.
In case it's not obvious from what others have said, doing QM on observables with discrete spectra looks very different from that on continuous spectra. Different mathematical tools are helpful in each case.
Scott Aaronson has an insightful essay on this in chapter 9 of his excellent book, "Quantum Computing Since Democritus". The TL;DR is that quantum mechanics can be derived as a generalization of probability theory using the 2-norm instead of the 1-norm. In Aaronson's words:
"Quantum mechanics is what you would inevitably come up with if you started with probability theory, and then said, let's try to generalize it so that the numbers we used to call 'probabilities' can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn't, but it could have been."
Well worth a read. In fact, I'd say it's worth buying the book just for this one chapter.
To counter your enthusiasm I must say that I rather disliked his reasoning. My problem is essentially that what Aaronson's calls "the theory" is a somewhat bastardized version of quantum mechanics that might suffice for quantum computing but, in my opinion, not for physics.
I think this is a very common view, and a somewhat mistaken one.
Quantum mechanics is used in very different ways by people in quantum information, condensed matter, many body physics, quantum field theory, nuclear physics and may other (sub)fields.
Of course it will be difficult for a quantum information theorist if they try to apply what they know directly to a hydrogen atom, but (speaking from experience) it will also be quite difficult for someone trained in what you call Hermitian quantum mechanics to directly apply what they know to quantum field theory, or quantum information or any other subfield that uses different language.
I strongly disagree with your summary "if you know Hermitian quantum mechanics then unitary quantum mechanics is conceptually straightforward. If you know unitary quantum mechanics then you will have a lot of new concepts and mathematics to learn before you understand hermitian quantum mechanics".
I challenge anyone trained in Hermitian quantum mechanics to make progress on (for example) proving or disproving the generalised quantum Stein's lemma, or any of the unsolved problems here https://arxiv.org/abs/2002.03233 using those methods.
With conceptually straightforward I meant that the concepts are easy to pick up.
For example, the paper you cite is entirely understandable for anyone with some training in hermitian QM. In contrast, good luck trying to understand elementary concepts like the spectrum of the hydrogen atom or interference of matter waves from unitary QM.
Of course the field of quantum info has progressed enormously and has its own interesting challenges, for which hermitian QM is all but useless.
I think maybe we have some difference in how we're talking about things.
Concepts like the spectrum of the hydrogen atom or interference phenomena aren't particularly difficult to understand conceptually: the Hamiltonian has some eigenvectors and eigenvalues, you use the Dirac equation and work them out. The "matter waves" interfere essentially in the same way that waves on the surface of a pond do.
The things that you're calling conceptual understanding I guess must be different to this: maybe something like detailed calculations of the structure of the spectrum?
To counter your unenthusiasm, think about it from a mathematician's perspective. Mathematics of QM does not live in some separate corner created to do physics. The need for QM created short-lived confusion, now it's all embedded into a much larger coherent mathematical structure.
For a pure mathematician, quantum mechanics is a lovely introduction to Hilbert Spaces.
That's only kind of true. Standard continuous-variable quantum mechanics has all sorts of consistency problems, and the only way to get reasonable predictions out is to paper over infinities and pretend they don't exist.
I know there are what are called C*-algebras, which help solve some of these issues, but I don't know anything about them. I do know that the Hilbert Space approach is not sufficient.
True, but from a mathematician's point of view, the theory quickly becomes complicated (and in some sense limited) if you really want to do things rigorously when working with continuous systems, something that does not happen with finite dimensional systems as the parent comment probably alludes to.
I was thinking about Aaronson when I added the CHSH game to the comment, because one of his essays (which I unfortunately could not find anymore) made that click for me.
On yesterday’s live stream[0] on Stephen Wolfram’s Twitch the team went through several improvement proposals to functions in WolframAlpha, including QuantumCircuitOperator which is a variant of a String Diagram.
Before this I didn’t know Stephen hosted “Live CEOing” sessions and now I wish this was the norm!
Mindscape episode with Stephen Wolfram from 2021 which might also be of interest. I enjoyed it even if most of it was way over my head. It is probably overly-ambitious and probably a dead end but it's an worthy attempt (if you have the money) and he's definitely not a crank. Eccentric maybe though.
I just skimmed the article for sanity checking and it looks more like crackpottery than science to me.
Looking at the numbers on the graphs for single-slit diffraction, they are just binomial coefficient, at least mostly, not sure why there are pieces missing in the last rows. That is also what you expect when you repeatedly make binary decisions to go left or right. The article does not mention the binomial distributions once, it only appears in a comment.
And then they claim that it converge to the actual single-slit diffraction distribution, something with a Chebyshev polynomial and the sinc function, according to the article. Seemingly without justification besides looking at graphs and noting that they are both bell shaped. As said, not sure what is going on in the last rows of the graphs, but I would almost bet that the two functions are not the same, even in the limit as it becomes a Poisson distribution plus whatever the last rows do.
Why do they not just proof that the two are the same? The entire article seems to be about getting numbers out of their multiway system and then concluding that - if you squint hard enough - they look somewhat like diffraction patterns.
> I would almost bet that the two functions are not the same, even in the limit as it becomes a Poisson distribution plus whatever the last rows do.
A Gaussian distribution, I think. But they're certaintly not the same function, and it should be immediately obvious to a math grad with experience in physics. The sinc function, for one, has secondary maxima (its plot in the article is very convenienty cropped to allow pretending those don't exist). Just put a hair in the path of a laser beam and you will see the local maxima in light intensity! Their "single-slit" string procedure, on the other hand, can only generate a single central peak. This really makes no sense at all.
> Regarding the similarity to a double slit interference pattern, what I found in my investigation (see my previous comment) is it was solely the result of the sorting algorithm used for the X coordinates of each possible outcome/weight. Ordering by the average X location results in a binomial distribution, but there are still ties to deal with. Applying secondary sorting to those based on the position of the left or rightmost X creates secondary binomial distributions within the main binomial distribution, similar to what Jonathan showed, except without all the spikiness because his example wasn’t consistently sorted in this manner.
> You can see the sorting I’m talking about by looking at the sorting of Xs in the output of weights in Jonathan’s double-slit examples. The bottom half is sorted as I stated, but the top half is somewhat out of order, corresponding to the spikes in the tallest curve in the center of the graphs.
> Maxime C, I had the same question as you – Why choose this sorting method for encoding the photon positions? My conclusion is that it doesn’t work however. Each additional step run with a large string produces an additional secondary binomial distribution. To get a smooth curve, we’d need to run billions of steps on a large string, but then we we’d also see billions of peaks, which no longer compares well with the expected diffraction pattern.