- Reaction score
- 1,697
Two researchers have proved that Penrose tilings, famous patterns that never repeat, are mathematically equivalent to a kind of quantum error correction.
If you want to tile a bathroom floor, square tiles are the simplest option — they fit together without any gaps in a grid pattern that can continue indefinitely. That square grid has a property shared by many other tilings: Shift the whole grid over by a fixed amount, and the resulting pattern is indistinguishable from the original. But to many mathematicians, such “periodic” tilings are boring. If you’ve seen one small patch, you’ve seen it all.
In the 1960s, mathematicians began to study “aperiodic” tile sets with far richer behavior. Perhaps the most famous is a pair of diamond-shaped tiles discovered in the 1970s by the polymathic physicist and future Nobel laureate Roger Penrose. Copies of these two tiles can form infinitely many different patterns that go on forever, called Penrose tilings. Yet no matter how you arrange the tiles, you’ll never get a periodic repeating pattern.
“These are tilings that shouldn’t really exist,” said Nikolas Breuckmann, a physicist at the University of Bristol.
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.
www.quantamagazine.org
If you want to tile a bathroom floor, square tiles are the simplest option — they fit together without any gaps in a grid pattern that can continue indefinitely. That square grid has a property shared by many other tilings: Shift the whole grid over by a fixed amount, and the resulting pattern is indistinguishable from the original. But to many mathematicians, such “periodic” tilings are boring. If you’ve seen one small patch, you’ve seen it all.
In the 1960s, mathematicians began to study “aperiodic” tile sets with far richer behavior. Perhaps the most famous is a pair of diamond-shaped tiles discovered in the 1970s by the polymathic physicist and future Nobel laureate Roger Penrose. Copies of these two tiles can form infinitely many different patterns that go on forever, called Penrose tilings. Yet no matter how you arrange the tiles, you’ll never get a periodic repeating pattern.
“These are tilings that shouldn’t really exist,” said Nikolas Breuckmann, a physicist at the University of Bristol.
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.
Never-Repeating Tiles Can Safeguard Quantum Information | Quanta Magazine
Two researchers have proved that Penrose tilings, famous patterns that never repeat, are mathematically equivalent to a kind of quantum error correction.
www.quantamagazine.org
