Accepted submission by **JoeMerchant **
at 2024-03-23 16:10:01
**from the ****Aperiodic indistinguishability** dept.

For over half a century, aperiodic tilings have fascinated mathematicians, hobbyists and researchers in many other fields. Now, two physicists have discovered a connection [quantamagazine.org] 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 [aps.org] 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.

-----

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 mathemeticians less than half my age...