Fractal "Matrix" and Computational Uncertainty Principle

In 2017 two physicists: Zohar Ringel and Dmitry Kovrizhin published an article:

Quantized gravitational responses, the sign problem, and quantum complexity

here is a popular science summary in Cosmos Magazine:

Physicists find we’re not living in a computer simulation

The article "proves" we do not live in some Matrix because they found something that will be difficult for the Matrix to calculate.

What I'm writing here started as my refutation to this article but grew up to be a new and original twist on the old theory of "Simulation Hypothesis" plus an explanation to Heisenberg's uncertainty principle, that until now was unexplained.

Heisenberg's uncertainty principle

Heisenberg proved that even theoretically with infinitely accurate tools, we can never measure both the location and the momentum (for our purpose "speed") of a particle. The complete list of paired properties that Niels Bohr considered can be found in Wikipedia under Complementarity (physics) :

Why the "no Matrix" article is wrong

The article claims that if we live in a simulated reality that runs on a conventional computer (not a quantum computer) let's say by aliens, and if we inside this reality will build a quantum computer, then the aliens' computer will not be able to calculate the calculations of our quantum computer.

The article says that although we have tried for years to simulate quantic processes on coventional computer, we failed. and this is because one of two reasons: Either we lack the analytic sophistication, or there is a physical barrier to do this.

The article says that creating a barrier in a conventional computer is not well defined, so they will create a barrier in a computer that operates in the Quantum Monte Carlo method which involves some mathematical problem with the sign.

Of course we can solve this immediately in saying that the aliens also use quantum computer or something better that we don't know. Even if we (unreasonably) assume that we invented the quantum computer before the aliens, the moment they see that we invented it, it will overload their coventional computer cpu, and in a very short time they will build a quantum computer of their own and copy us there without us feeling anything.

This is also mentioned at the end of the article in Cosmos Magazine:

There is a caveat to this conclusion: if our universe is a simulation, there is no reason that the laws of physics should apply outside it. In the words of Zohar Ringel, the lead author of the paper, “Who knows what are the computing capabilities of whatever simulates us?”

But if we stick with the assumption that the aliens use a conventional computer, we reach something a lot more exciting. First of all we don't need to go to the quantum world to find something that we can't compute.

The vast majority of equations in the real world are non linear equations, and almost all of them we can't solve. A classic example is the "Three-Body Problem" (like Moon, Earth, and Sun), that Poincare first tried to solve, and we still can't solve it accurately today. This was the first time in history that humanity ran into "Chaos Theory".

Calculating the gravitation between two bodies and how they rotate each other in space we can do accurately since Newton. But if you add another body to the system and it turns chaotic, and we can not accurately calculate this. But we can numerically compute to a close approximation for a specific point in space what will be the effect there. And this also with great effort.

In the article they also largely rely on a process which when you add one more particle to the system, the number of possibilities grows exponentially. That process is based on the Hall Effect.

But although there is a wonderful explanation about the "Ordinary Hall Effect" in "Sixty Symbols"

And a nice interview about the "Quantum Hall Effect" by Steven Girvin

We will not go into the Hall Effect, because we will think instead about chaos in general and fractals like the "Mandelbrot Set" in particular.

By the way what is created in the three body problem is a Newton Fractal that you can see in Wikipedia and a prettier version in the book "Chaos: Making a New Science" by James Gleick (called THE COMPLEX BOUNDARIES OF NEWTON’S METHOD in the color pictures section).

So if the problem is a chaotic system, it makes sense to try a solution that helps in another chaotic system - fractals in computer graphics.

If we look at a picture of a fractal that we created in our computer (for example: Mandelbrot set), there is an infinite amount of details, and the deeper you go the more details we see.

Does that mean that we need an infinite computer to draw this fractal from a "Bird's-eye view"? No, we just need the right formula. For the Mandelbrot Set it's the equation f(z)=z^2+c All the endless beauty of all the "zoom" movies are inside this short and elegant formula. Even if you don't like math, you understand it's a short formula; And even if you don't know much about computers, you understand that the fact that it's drawn every day on personal computers, means that you don't need infinite computing power.

So for example if the above article's authors build a quantum computer that calculates something equivalent to building "Koch Snowflake",

This looks like it should take forever to draw, but since we only have to draw it up to the current limited resolution,

The aliens' conventional computer (or even a human personal computer) can draw it easily.

And if they want to zoom (enlarge) into a deeper and deeper specific part,

Then that conventional computer only has to work on a smaller and smaller part of the picture, and can easily do it.

Reality built from fractals

(this tiny amazing demo by Tim Clarke is only 6 KiloByte! (about half the size of this text page that you are reading now), and runs perfectly smooth on an old 486 computer, and yet generates a whole interactive martian planet surface - based on a fractal! by the way I was the one who uploaded it first to website - my nickname there was Chompi)

My theory is that the aliens draw our reality exactly like fractals, and all our physical laws of nature that we found are only a rough approximation to the elegant equations (that unfortunately we don't know) of the aliens. Our knowledge in this metaphor is like the "Feigenbaum Constants" - like the ratio between the circles of the Mandelbrot Set. which is close to 4.669...

So this falls into the category of "we lack the analytic sophistication", that the article authors have considered.

The "proof" I found to the fact that we are indeed inside a conventional computer (not infinitely powerful) comes from Heisenberg's uncertainty principle in Quantum Mechanics.

Uncertainty: Standard explanation vs Fractal explanation

Standard Explanation - Bottom-Up Calculation

The effect comes from the wave nature of the particle. You can read it in "Wave - Particle Duality" in Wikipedia

Fractal Explanation - Top-Down Calculation

When I was a teenager (when Tim Clarke created his amazing Mars Demo), the personal computer graphics card was limited for exmple by two parameters: number of colors, and screen resolution.

We could use the limited memory of the graphics card (VGA) :

EITHER for more colors and less resolution

OR for more resolution and less colors.

There was a tradeoff (barter trade or compromise) between horizontal resolution and number of colors; There was also a tradeoff between vertical resolution and screen refresh rate (how many times in a second the screen is redrawn).

Let's say it's the same thing in our reality. Then if we know exactly the position of the particle (resolution) we know less about it's speed (the ratio between screen refresh and it's progress between the pixels).

The next pair is energy and time. You can't know exactly how much energy a particle has and also when exactly it left the system (see Einstein's Box).

If you think of color as energy, in the same way that red light rays have lower frequency (and so also lower energy) than blue-purple light rays, and if we think of screen refresh rate as time, then there is a tradeoff corelation between this two properties, because of the link between horizontal resolution and vertical resolution.

What about the property pair of spin in different directions?

This is similar to something in VGA graphics called mode-X invented by the legendary programmer Michael Abrash. Mode-X lets you "unfold" the "folded" memory of the VGA card and treat it as a planar memory. Then you can define a few pages of memory off the screen EITHER to the sides OR up and down, and flip between them for smooth animation.

If we treat these ways of unfolding as the frames of the rotation, let's say we divided the memory into 4 horizontal spread pages and in each one the object is rotated a quarter of a rotation in a certain direaction - let's say horizontal rotation. Of course we can't use it in the same time for rotation around another axis - let's say vertical rotation.


When I found out that there is a book called:

Programming the Universe: A Quantum Computer Scientist Takes On the Cosmos

by Prof. Seth Lloyd who calls himself a "quantum mechanic".

I was worried that I'm just re-inventing the wheel here, repeating what someone else already though about.

But I read the section about the Heisenberg's uncertainty principle, and the explanation is not similar to mine.
As far as I can understand his book talks about the universe running on a quantum computer (not conventional).