4 ms·

Never-Repeating Tiles Can Safeguard Quantum Information

98 points by digital55 3y ago

infogulch 3y ago
> it’s impossible to tell any two tilings apart by examining any local area. That’s because every finite patch of any tiling, no matter how large, will show up somewhere in every other tiling I was about to pose a question but it turns out that's what the article is about!
_fw 3y ago
love how that piece quotes Tony Cubitt, a stunning coincidence for somebody who works in quantum computing-related science. His website is a fun read too: https://www.dr-qubit.org/qubit.html https://www.dr-qubit.org/qubit.html
- nabla9 3y ago
  https://en.wikipedia.org/wiki/Nominative_determinism https://en.wikipedia.org/wiki/Nominative_determinism
- andyferris 3y ago
  Yes, I too had a chuckle many years ago when I first became aware of Tony’s work in QIT.
link0ff 3y ago
Much simpler would be just to use Rule 30.
- snarkconjecture 3y ago
  I have to admit I don't immediately see how that would work, since the proof uses properties of Penrose tilings other than aperiodicity, but if you flesh out the details I think it would be a great paper.
jakedata 3y ago
A linked article about an einstein (one-piece) aperiodic tile contains a beautifully simple animation of the continuum of tiles that morph as you scroll down the page. https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/ https://www.quantamagazine.org/hobbyist-finds-maths-elusive-...
lupire 3y ago
> edges of tilings don’t fall at regular intervals, so specifying their distribution requires continuous real numbers rather than discrete integers. Surely they mean algebraic numbers not real numbers.
- ysofunny 3y ago
  > Surely they mean algebraic numbers not real numbers. that's an extremely subtle distinction! I don't really understand this distinction to my own satisfaction. in my mind the concept I need to understand better is the generalized notion of 'decimal point' this presents me 2 avenues of further investigation: 'decimal' which just means ten which is already known as the general concept of the base of a positional numeral system (so this links to exponents and modulo arithmetic and the modern algebraic framework) and the other avenue is the 'point'. so here's my question: up to which extent should I attempt to realize the notion of "fixed point" (like in Tarski's theorem) and the specific application of a "fractional"-point as an instance (where 'decimal' is the specific applied case n=10=2*5) of a "fixpoint" used as part of a positional numeral value determination system?
  - snarkconjecture 3y ago
    A decimal "point" is called that because it's written as a dot: . "Fixed point" arithmetic has nothing to do with "fixed point theorems", it's a name collision. Base ten has nothing to do with the distinction between real and algebraic numbers. If you Google "algebraic vs transcendental number" you'll find many good resources. Pi is an example of a transcendental number.
  - dhosek 3y ago
    At the risk of telling you something you don’t already know: algebraic numbers are an extension of integers¹ which adds in those numbers which are zeros of finite polynomials with integer coefficients.² So, to pick a trivial example, ±√2 are algebraic number because they are solutions to x²-2=0. But there are many real numbers which are not algebraic. π and e are the best known non-algebraic numbers (we call these numbers transcendental. In common parlance, unless otherwise specified, the term transcendental number is generally used to refer to the set ℝ\ where refers to the set of algebraic numbers in this instance³). In fact, statistically speaking, the probability that any given real is algebraic is 0.⁴ ⸻ 1. It’s worth noting that while it’s easy to assume that algebraic numbers are a subset of the reals, the fact that it includes complex numbers (i is algebraic since it’s a solution to x²+1=0) means it’s not. n.b., just as not all reals are not algebraic, not all complex numbers are algebraic. 2. Rational coefficients of a polynomial can be trivially eliminated by simply multiplying all terms of the polynomial by the lcm of the denominators of any non-integer polynomial, so the set of zeros of polynomials with integer coefficients and the set of zeros of polynomials with rational coefficients will be identical. 3. I’m aware is also often used to label the adele ring, but I’m happy to borrow it for the purposes of a Hacker News comment. 4. The proof of this statement is left as an exercise to the reader.

98 points by digital55 3y ago

> it’s impossible to tell any two tilings apart by examining any local area. That’s because every finite patch of any tiling, no matter how large, will show up somewhere in every other tiling I was about to pose a question but it turns out that's what the article is about!

_fw 3y ago

love how that piece quotes Tony Cubitt, a stunning coincidence for somebody who works in quantum computing-related science. His website is a fun read too: https://www.dr-qubit.org/qubit.html https://www.dr-qubit.org/qubit.html

nabla9 3y ago
https://en.wikipedia.org/wiki/Nominative_determinism https://en.wikipedia.org/wiki/Nominative_determinism
andyferris 3y ago
Yes, I too had a chuckle many years ago when I first became aware of Tony’s work in QIT.

link0ff 3y ago

Much simpler would be just to use Rule 30.

snarkconjecture 3y ago
I have to admit I don't immediately see how that would work, since the proof uses properties of Penrose tilings other than aperiodicity, but if you flesh out the details I think it would be a great paper.

jakedata 3y ago

A linked article about an einstein (one-piece) aperiodic tile contains a beautifully simple animation of the continuum of tiles that morph as you scroll down the page. https://www.quantamagazine.org/hobbyist-finds-maths-elusive-einstein-tile-20230404/ https://www.quantamagazine.org/hobbyist-finds-maths-elusive-...

lupire 3y ago

> edges of tilings don’t fall at regular intervals, so specifying their distribution requires continuous real numbers rather than discrete integers. Surely they mean algebraic numbers not real numbers.

ysofunny 3y ago
> Surely they mean algebraic numbers not real numbers. that's an extremely subtle distinction! I don't really understand this distinction to my own satisfaction. in my mind the concept I need to understand better is the generalized notion of 'decimal point' this presents me 2 avenues of further investigation: 'decimal' which just means ten which is already known as the general concept of the base of a positional numeral system (so this links to exponents and modulo arithmetic and the modern algebraic framework) and the other avenue is the 'point'. so here's my question: up to which extent should I attempt to realize the notion of "fixed point" (like in Tarski's theorem) and the specific application of a "fractional"-point as an instance (where 'decimal' is the specific applied case n=10=2*5) of a "fixpoint" used as part of a positional numeral value determination system?
- snarkconjecture 3y ago
  A decimal "point" is called that because it's written as a dot: . "Fixed point" arithmetic has nothing to do with "fixed point theorems", it's a name collision. Base ten has nothing to do with the distinction between real and algebraic numbers. If you Google "algebraic vs transcendental number" you'll find many good resources. Pi is an example of a transcendental number.
- dhosek 3y ago
  At the risk of telling you something you don’t already know: algebraic numbers are an extension of integers¹ which adds in those numbers which are zeros of finite polynomials with integer coefficients.² So, to pick a trivial example, ±√2 are algebraic number because they are solutions to x²-2=0. But there are many real numbers which are not algebraic. π and e are the best known non-algebraic numbers (we call these numbers transcendental. In common parlance, unless otherwise specified, the term transcendental number is generally used to refer to the set ℝ\ where refers to the set of algebraic numbers in this instance³). In fact, statistically speaking, the probability that any given real is algebraic is 0.⁴ ⸻ 1. It’s worth noting that while it’s easy to assume that algebraic numbers are a subset of the reals, the fact that it includes complex numbers (i is algebraic since it’s a solution to x²+1=0) means it’s not. n.b., just as not all reals are not algebraic, not all complex numbers are algebraic. 2. Rational coefficients of a polynomial can be trivially eliminated by simply multiplying all terms of the polynomial by the lcm of the denominators of any non-integer polynomial, so the set of zeros of polynomials with integer coefficients and the set of zeros of polynomials with rational coefficients will be identical. 3. I’m aware is also often used to label the adele ring, but I’m happy to borrow it for the purposes of a Hacker News comment. 4. The proof of this statement is left as an exercise to the reader.