Never-Repeating Patterns of Tiles Can Safeguard Quantum Information
For over half a century, aperiodic tilings have fascinated mathematicians, hobbyists and researchers in many other fields. Now, two physicists have discovered a connection between aperiodic tilings and a seemingly unrelated branch of computer science: the study of how future quantum computers can encode information to shield it from errors. In a paper posted to the preprint server arxiv.org in November, the researchers showed how to transform Penrose tilings into an entirely new type of quantum error-correcting code. They also constructed similar codes based on two other kinds of aperiodic tiling.
At the heart of the correspondence is a simple observation: In both aperiodic tilings and quantum error-correcting codes, learning about a small part of a large system reveals nothing about the system as a whole.
...in 1995, the applied mathematician Peter Shor discovered a clever way to store quantum information. His encoding had two key properties. First, it could tolerate errors that only affected individual qubits. Second, it came with a procedure for correcting errors as they occurred, preventing them from piling up and derailing a computation. Shor's discovery was the first example of a quantum error-correcting code, and its two key properties are the defining features of all such codes.
...An infinite two-dimensional plane covered with Penrose tiles, like a grid of qubits, can be described using the mathematical framework of quantum physics: The quantum states are specific tilings instead of 0s and 1s. An error simply deletes a single patch of the tiling pattern, the way certain errors in qubit arrays wipe out the state of every qubit in a small cluster.
The next step was to identify tiling configurations that wouldn't be affected by localized errors, like the virtual qubit states in ordinary quantum error-correcting codes. The solution, as in an ordinary code, was to use superpositions. A carefully chosen superposition of Penrose tilings is akin to a bathroom tile arrangement proposed by the world's most indecisive interior decorator. Even if a piece of that jumbled blueprint is missing, it won't betray any information about the overall floor plan.
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I wonder... does the Penrose / Einstein principle of non-repetition preclude cylindrical, or even donut surface topology full tilings? If not, that could solve the infinite plane mapping into a physically realizable quantum computer problem. I do wonder, but not enough to give up all my other work and hobbies to pursue deeply theoretical mathematics being heavily studied by thousands of PhD mathematicians less than half my age...
Related:
Hobbyist Finds Math's Elusive 'Einstein' Tile
How Space and Time Could Be a Quantum Error-Correcting Code
How Quantum Computers Will Correct Their Errors
(Score: 2, Interesting) by pTamok on Tuesday March 26 2024, @09:54AM (3 children)
It's interesting, but I don't know enough to comment with any insight, even superficial.
I wish more people would be honest/humble enough to admit when they don't know. It would make some interactions much less fraught.
Tilings on surfaces with positive or negative curvature, or doughnuts (or, in fact, any non-self-intersecting surface e.g. a Klein bottle* in 4-space) sound interesting.
*In fact, thinking about it, I'm not even sure it is possible to aperiodically tile a Klein bottle in 4-space, but I can't express my thoughts formally.
(Score: 0) by Anonymous Coward on Tuesday March 26 2024, @01:25PM
Socrates called.
(Score: 2) by JoeMerchant on Tuesday March 26 2024, @02:00PM (1 child)
>*In fact, thinking about it, I'm not even sure it is possible to aperiodically tile a Klein bottle in 4-space, but I can't express my thoughts formally.
I prefer not to think about Klein bottles in 4 space, but... I get the same feeling about non-self-intersecting 2D surfaces: if you can close and completely tile the surface, a complete tiling (of a simple symmetric surface like a doughnut) is then de-facto periodic, so it at least _feels_ impossible, but wouldn't it be cool to demonstrate a way that it _is_ possible, like: deformations of a cylinder or doughnut that make a complete tiling possible, said deformations somehow destroying the periodicity... The cylinder feels like it would be a never ending problem of how to achieve the deformations for complete tiling around the circumference as you travel along the axis, but a deformed doughnut could eventually be "solved."
The thing I like about tiling studies is the physicality of it: you can print a sheet of tile patterns, cut them out, physically lay them out (on a 2D surface) and readily demonstrate what's going on. Then, if you're one for self-abuse, you can take the concepts up into 3D, and beyond.
Now, as for quantum superposition - I never have gotten the warm fuzzies about superposition. I can't get past the idea that hidden variables explains it, and that leads to superdeterminism [wikipedia.org]. Folding up a tiny 2D universe into a doughnut kind of sums up my feelings on the matter. Then there's the fact that the whole field hangs their hat on Bell's Theorem which is based entirely upon statistical analysis of assumptions, and I am a strong adherent of the "1) Lies 2) Damn Lies 3) Statistics" camp.
Of course, there's a whole field of quantum computationalists out there building big expensive devices which may, someday, start to solve problems faster than conventional computers... wouldn't it be a hoot if everybody is right: superdeterminism is "a thing" and quantum computers are just revealing truths that the universe already knows, but we're just barely beginning to discover how to ask it?
As for the doughnuts - I always liked running 2D simulation maps on a donut topology, it seems so much more elegant than having "hard walls" at the edges of the simulated universe. I've done rectangular tiles, and hex tiles, but distorting an aperiodic tiling into a wrap-around donut would be an interesting twist on the genre. Useful for what, I'm not sure... maybe when scaled up it's closer to the actual quantum fabric underlying our Universe than anything we've imagined so far.
🌻🌻 [google.com]
(Score: 2) by JoeMerchant on Tuesday March 26 2024, @02:06PM
P.S. - aperiodic tilings of something resembling a distorted sphere may also be quite interesting... The tile side lengths can be expressed in arc-seconds with the tiles mapped to a surface, possibly of varying radius...?
And, on the subject of arc-measurements: there are real physical phenomena that map nicely into magnitude-direction vectors, where the direction is a 0-2pi wrapping measurement. In the signals measured by MRI, changes in phase of the MRI signal correspond very linearly with changes in temperature of the subject being imaged.
🌻🌻 [google.com]