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"The best way to understand their approach is by considering something else ordered yet non-repeating: "quasicrystals." A typical crystal has a regular, repeating structure, like the hexagons in a honeycomb. A quasicrystal still has order, but its patterns never repeat. (Penrose tiling is one example of this.) Even more mind-boggling is that quasicrystals are crystals from higher dimensions projected, or squished down, into lower dimensions. Those higher dimensions can even be beyond physical space's three dimensions: A 2D Penrose tiling, for instance, is a projected slice of a 5-D lattice.

For the qubits, Dumitrescu, Vasseur and Potter proposed in 2018 the creation of a quasicrystal in time rather than space. Whereas a periodic laser pulse would alternate (A, B, A, B, A, B, etc.), the researchers created a quasi-periodic laser-pulse regimen based on the Fibonacci sequence. In such a sequence, each part of the sequence is the sum of the two previous parts (A, AB, ABA, ABAAB, ABAABABA, etc.). This arrangement, just like a quasicrystal, is ordered without repeating. And, akin to a quasicrystal, it's a 2D pattern squashed into a single dimension. That dimensional flattening theoretically results in two time symmetries instead of just one: The system essentially gets a bonus symmetry from a nonexistent extra time dimension."

7 comments

Feeling resigned to just not understanding this one as a lay person. Oh well, hope it leads to more cool things!

Ok, so let's cut through the woo:

A crystal is a repeating pattern of elements in space. For example, a diamond is carbon atoms - the same thing in ordinary coal - arranged in a particular shape of grid.

You can have patterns that are made in time rather than space, such as by hitting a drum with a stick in time with music. Of course, this isn't really very crystal-like, because the drum doesn't try to resist you hitting it off-time. However, there are certain atomic-scale materials that do resist your horrible off-beat drumming, and you "hit" them with a laser rather than a drumstick. These systems are time crystals[0].

You can also have crystal patterns that don't repeat, which are called quasicrystals. For every quasicrystal, there's a higher-dimension crystal that it is a shadow of. You could imagine, say, a 3D grid or lattice that you can shine a light through onto a piece of paper to get an irregular 2D pattern, which would be your quasicrystal. The two structures are related to one another, but that doesn't necessarily mean that the flatlanders living in it have proof of the existence of a third dimension.

The new development is time quasicrystals: i.e. a drum that you can bang with some non-repeating pattern and it will also keep in time with the pattern even if you are off. The stuff about "acting like it has two time dimensions" is more woo; there is a 2D time relation to the 1D time quasicrystal, but there is no actual 2D time shenanigans going on. The non-repeating pattern apparently also makes the time crystal better at "keeping time" which may help build more stable qubits for quantum computers.

[0] Note that you can't have spacetime crystals in the same material. You can either have atoms that link to one another with chemical bonds to form a pattern, or atoms that trade their bonds in rhythmic patterns, but not both.

> You could imagine, say, a 3D grid or lattice that you can shine a light through onto a piece of paper to get an irregular 2D pattern, which would be your quasicrystal.

This is where I lose it. I actually can't imagine such a thing. Every regular 3D crystal I imagine has a repeating pattern in its shadow. For every ray of light passing through one part of the 3D lattice, I can locate parallel rays that produce the same result in other parts of the lattice.

What am I missing here? Just not imagining the right lattice types? Or are we assuming a point-source of light so that no 2 rays are parallel?

Look at this image: https://i.pinimg.com/736x/54/1b/1a/541b1afd4a72564f808825b3e...

The window is the lattice, which is regularly ordered. The shadow, however, is distorted, ie each light beam is not the same size as the one next to it.

... but that window is a 2D lattice, with a 2D shadow.

> For every quasicrystal, there's a higher-dimension crystal that it is a shadow of.

So what's the 3d crystal whose shadow is the Penrose tiling? The article says it's a "projected slice of a 5D lattice", which I really struggle to visualize.

Or perhaps easier, what's the regular 2D pattern of which the Fibonacci sequence is a projection?

The system producing the shadow isn't a 2d lattice, because it also involves the sun. Changing the location of the sun relative to the window will change the pattern of the shadow.

If you wanted to really convince yourself, add multiple windows such that the shadow is affected by the 'swiss cheese' effect of the holes of all the windows lining up relative to the sun.

For this fibonacci sequence, consider a 2d grid with Vertical and Horizontal lines. Take a line with slope 1/golden ratio, and run it through this grid. If you mark down H for every Horizontal line you cross, and V for every Vertical line you cross, you get this fibonacci sequence (properly called a fibonacci word).

This is the relationship between a 2d lattice and this sequence that wikipedia told me. Calling it a projection seems a bit much to me. Expanding on this, I would expect the Penrose tilings to be a similar slice through a regular high-dimensonal 'crystal'. The key being that irrationally of the slope means no periodicity of the intersections.

The window is not 2D, it's 3D with depth as well. Now imagine that but with even more detail, like a cube lattice.

Just imagine 2 transparent sheets with some regular pattern - when you move one over the other you can create infinite, non-repeating patterns.

You're struggling to visualize this? "You're just not thinking fourth dimensionally." --Doc Emmet Brown (https://www.youtube.com/watch?v=CUcNM7OsdsY)

A projection of the 2D square lattice can produce the binary fibbonacci sequence.

This picture I stole from some blog shows this projection: https://grahamshawcross.files.wordpress.com/2012/04/spots.jp...

A 2D square lattice is defined by two perpendicular basis vectors denoting nearest-neighbor distances. Lets put a grid point at the origin at define position as (a,b), where a and b are in units of the basis vectors. That is every integer (a,b) is a grid point that is a hops to the right and b hops up from the origin. Lets also define directions [a,b], where this is the vector from the origin (0,0) to point (a,b).

Consider the following operation: we draw an arbitrary line on the lattice then take all the points within some distance of that line and project them onto the line.

If you draw the line parallel to either basis vector, i.e. [1,0] or [0,1], you will get a 1D sequence where every point is identical: a 1D grid. This is actually independent of the size of the neighborhood around the line we consider as long as it is large enough to include any points; the projections of more distant points align with closer points due to the symmetry of the lattice. A line at 45 degrees (i.e. [1,1]) produces a similar result.

What about some other integer vector? If we draw a line along [5,7], the projected points will no longer be as tidy, but after some distance along the line, we will reach a point equivalent to where we stared: the grid point (5,7). And then (10,14) and (15,21) so on. Every time we hit a new grid point, the pattern of the projection will repeat. The specific pattern between grid points may vary as we change the size of the neighborhood around the line we consider for the projection, but it will always retain the same periodicity. Like the previous cases, as we increase neighborhood size, the projection will stop changing after some critical value as all new points in the neighborhood will line up with previous points. You can see this properties by playing with a piece of graph paper. All rational vectors have parallel integer vectors, and so will have some underlying periodicity.

What about an irrational vector, say [e,pi]? After leaving the origin travelling on this path your will ~never~ hit another grid point. Therefore the projections of grid points will be aperiodic. Not only that, as we increase the neighborhood size, the pattern constantly changes: each new point always has a new spot on the projection. However despite being aperiodic, the system is still ordered: you can know where the next point will show up every time.

It turns out, if you draw a line along the vector [phi,1], where phi is the golden ratio, and use a neighborhood size of sqrt(2), the projection of the points has another interesting property: there are only two possible distances between projected points on the line, lets call them long (L) and short (S), themselves related by the golden ratio. The pattern of Ls and Ss is itself ordered and aperiodic. Not only that, but a simple substituion relation L->LS and S->L produces a longer sequence that includes the original. This just happens to be the substitution relation for finding subsequent terms in the binary fibbonaci sequence.

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I am sorry for this long-winded explanaton, but I am hoping it helps with visualization without pictures. Perhaps I should write a short blog post with some pictures. The key points are: starting from the origin a rational vector on a grid will always hit another grid point (and then infinite additonal points), which will define periodic relationships will all other grid points to the line. An irrational vector will never hit another grid point, so the relationships are always aperiodic. For a specific choice of projection and vector, you can recover the binary fibbonacci sequence.

If you angle the 3D lattice at an irrational angle relative to the 2D plane the points will not be perfectly periodic in the plane.

What is an irrational angle? Is this something I can actually do physically, or is it more of a theoretical math thing? For example, if I'm holding a toy that is a lattice showing the 3d structure of carbon between my dining room table and ceiling lamp, how do I rotate it such that it is irrational relative to my table?

this might help a little: https://math.stackexchange.com/questions/791848/penrose-tili... which is just a watered down explanation of the excellent https://tilings.math.uni-bielefeld.de/substitution/fibonacci...

I'm guessing it's irrational as in rational vs irrational numbers. Rational means a fraction of whole numbers, so irrational numbers are those which cannot be represented as such a fraction. A 1/4 turn is rational, a 1/pi turn is irrational.

I feel like the light has to be parallel for it to work, so sunlight is a better example than a table lamp. Although I can't imagine any rotation of a simple 3D lattice having a nonrepeating shadow. Perhaps a more complex 3D crystal is necessary?

Rational/ irrational here depends on the unit of measurement. A full circle (360 degrees) is rational if you measure it in degrees, but irrational if you measure it in radians (it's 2 pi radians).

You need an irrational 'slope'. Like sqrt(2), or pi

I had the same confusion as you, but I'm going to take a guess that it might be analogous to the following:

The function n -> sin(n) might be called a "shadow" of t -> sin(t), where k is an integer and t is a real number: namely, it's not periodic, but it's a shadow (projection from reals to integers) of something periodic.

Maybe someone can confirm if the analogy is correct here?

This video gives a very concrete example of a non-repeating pattern (Penrose tiling) https://www.youtube.com/watch?v=48sCx-wBs34

I understand the non-repeating patterns. I just don't see how a regular 3D lattice can produce such a pattern. Unless the light source creating this shadow is a point-source rather than a parallel one?

I guess I'm just looking for confirmation on this thought: A parallel light shone through a repeating 3D lattice will always produce a repeating 2D lattice.

Try projecting it on a surface with curvature. The projected grid spacing will be irregular and follow the curvature.

My guess is that it has to do with projections at an 'irrational' slope. That would prevent repetition, though I believe it would cause a dense set of points if you project the infinite lattice to a lower dimension.

> there are certain atomic-scale materials that do resist your horrible off-beat drumming

whoa.

Thanks this was an excellent explanation and even though I don't really understand, I feel like I do.

If you look at the graphic at the top of the article (Penrose tiling) you'll notice there are a bunch of points that are centers of rotational symmetry (you can rotate it 2pi/N and get the same thing) and lines of reflection symmetry (you can mirror it over that line and get the same thing) but there is no translational symmetry (you can't slide it over in any direction and overlap with the original), this is a "quasicrystal" (in 2d)

Compare to e.g. a grid of squares that has reflection and rotation symmetry but also has translational symmetry, this is a true "crystal" (in 2d)

This article is treating a train of laser pulses as a "1d crystal" and if long/short pulses resemble a Fibonacci sequence treating it as a "1d quasicrystal". This seems to be noteworthy in that using such a structured pulse train provides some improvements in quantum computing when it's used to read / write (i.e. shine on) information (i.e. electron configuration) from atoms / small molecules (i.e. qubits)

Edit: And the "2 time dimensions" thing is basically that a N-d "quasicrystal" is usually a pretty close approximation of an [N+M]-d "true crystal" projected down into N dimensions so the considering the higher dimension structure might make things easier by getting rid of transcendental numbers etc.

"Feeling resigned to just not understanding this one as a lay person."

The biggest and most important step is to make sure you drop any mysticism about what a "dimension" is. It's just a necessary component of identifying the location of something in some way. More than three "dimensions" is not just common but super common, to the point of mundanity. The location and orientation of a rigid object, a completely boring quantity, is six dimensional: three for space, three for the rotation. Add velocity in and it becomes 12 dimensional; the six previous and three each now for linear and rotational velocity. To understand "dimensions" you must purge ALL science fiction understanding and understand them not as exotic, but painfully mundane and boring. (They may measure something interesting, but that "interestingness" should be accounted to the thing being measured, not the "dimension". "Dimensions" are as boring as "inches" or "gallons".)

Next up, there is a very easy metaphor for us in the computing realm for the latest in QM and especially materials science. In our world, there is a certain way in which a "virtual machine" and a "machine" are hard to tell apart. A lot of things in the latest QM and materials science is building little virtual things that combine the existing simple QM primitives to build new systems. The simplest example of this sort of thing is a "hole". Holes do not "exist". They are where an electron is missing. But you can treat them as a virtual thing, and it can be difficult to tell whether or not that virtual thing is "real" or not, because it acts exactly like the "virtual" thing would if it were "real".

In this case, this system may mathematically behave like there is a second time dimension, and that's interesting, but it "just" "simulating" it. It creates a larger system out of smaller parts that happens to match that behavior, but it doesn't mean there's "really" a second time dimension.

The weird and whacky things you hear coming out of QM and materials science are composite things being assembled out of normal mundane components in ways that allow them to "simulate" being some other interesting system, except when you're "simulating" at this low, basic level it essentially is just the thing being "simulated". But there's not necessarily anything new going on; it's still electrons and protons and neutrons and such, just arranged in interesting ways, just as, in the end, Quake or Tetris is "just" an interesting arrangement of NAND gates. There's no upper limit to how "interestingly" things can be arranged, but there's less "new" than meets the eye.

Unfortunately, trying to understand this through science articles, which are still as addicted as ever to "woo woo" with the word dimensions and leaning in to the weirdness of QM and basically deliberately trying to instill mysticism at the incorrect level of the problem. (Personally, I still feel a lot of wonder about the world and enjoy learning more... but woo woo about what a "dimension" is is not the place for that.)

They could have just said "aperiodic laser pulses" are used. No need to introduce fantastical sounding terminology about multiple time dimensions, which seems to have been done quite deliberately.

This may be the most eye-opening and clarifying thing I've read about this domain in literally years. Thank you.

The connection back to the complexity chasm that exists between NAND gates and Quake is also fantastic because as a "traditional" software engineer, it makes perfect sense.

It's also good remembering that most of the "academic science" that underlies computers was established almost 100 years ago. But it took this long for us to get GTA Online.

Whatever advances arrive from these developments in Quantum computing may not see practical groundbreaking applications until we're all very old and decrepit.

It's still incredible to hear about. The fact that our modern "wireless" world exists on fundamentally the same physical primitives as a radio wave pulsing morse code bouncing it off the ionosphere 100 years ago is mindboggling.

Yeah, extra spatial dimensions might be common as grass in visualisations, but extra TIME dimensions... those are pretty unusual.

This is a really wonderful explanation that removes the woo from QM. As a non-scientist, I've spent a lot of time reading about QM and trying to understand stuff, and eventually get lost in hand-waviness about dimensions and vague references to Schrodinger and his boxes of semi-cats. Thanks!!

It's wrong to say that a quasi-crystal is crystal from a higher dimension. You apparently get a quasi-crystal if you project a higher dimensional crystal, which I guess is neat. But really they are just trying to hype up their own results.

So essentially this is somehow akin to a network with hypercube topology - it’s got a mathematical relationship to an extra dimension but there’s no physical extra dimension.

I don’t think anything is claimed about the existence or non existence of a corresponding physical dimension

"The system essentially gets a bonus symmetry from a nonexistent extra time dimension."

It's in the last sentence of the GP

Does this then count as a sort of "holographic time"? Encoding two dimensions on a single stream of time.

..and can two of them create a Time Cube?

https://knowyourmeme.com/memes/the-time-cube

4-day confirmed. We're all educated stupid now.

But that's saying that if you have something repeating in the same way along the X axis, you have two spacial dimensions. That's not the way most of us use "dimensions". (The math may work out for their usage to not be nonsense, but it's considerably less than a "real" extra time dimension.)

"There are no quasicrystals only quasi scientists" ~Linus Pauling

Will this affect the reliability and sustainability of quantum computing?

Is time is distorted space ???