⚛️ quantum physics

Quantum error-correcting codes from aperiodic monotiles: the Hat and the Spectre

This paper extends the construction of quantum error-correcting codes from Penrose tilings to the recently discovered Hat and Spectre aperiodic monotiles, demonstrating that these structures support robust quantum information protection and, uniquely, encode superselected classical bits corresponding to their long-range chirality or handedness depending on the gauged isometry group.

Original authors: Josep Batle, Adam Bednorz

Published 2026-07-20

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Josep Batle, Adam Bednorz

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to hide a secret message inside a giant, infinite mosaic. You want the message to be safe even if a burglar comes along and smashes a hole in the middle of the floor, erasing a chunk of the tiles. In the world of quantum physics, this is called "error correction." The goal is to store information in a way that is "locally hidden"—meaning if you look at just a small, finite piece of the mosaic, you can't tell what the whole picture is or what the secret is. It's like looking at a single grain of sand and trying to guess the shape of the entire beach; if the sand looks the same everywhere, you can't figure out the beach's shape from just one spot.

For a long time, scientists thought only very specific, complex patterns could do this. But a new discovery in 2023 changed the game: the "aperiodic monotile." This is a single, weirdly shaped tile that can cover an entire floor without ever repeating a pattern, and it does so in a way that forces the whole floor to be unique. The big question for physicists is: Can these new, single-tile shapes protect quantum secrets just as well as the famous, complex patterns used before? If they can, they might offer a simpler, more robust way to build the quantum computers of the future, which are notoriously fragile and prone to errors.

This paper takes two of these new "magic tiles"—the Hat and the Spectre—and tests them to see if they can act as guardians for quantum information. The authors, Josep Batle and Adam Bednorz, discover that these tiles do indeed create a shield against errors, but with a surprising twist: they don't just protect quantum data; they also store a hidden, classical secret that behaves like a switch.

The Magic Tiles: The Hat and the Spectre

To understand the experiment, you need to know the players. In 2023, researchers found a single shape called the Hat (a polykite made of eight smaller kites) that can tile a floor only if you use both its "left-handed" and "right-handed" versions. It's like a puzzle piece that only fits if you have both a left shoe and a right shoe in the pile. Shortly after, they found the Spectre, a strictly "chiral" tile. This one is like a left-handed glove that only fits with other left-handed gloves; it never needs a mirror image to fill the floor.

The paper asks: If we build a quantum code (a way to store data) using these tiles, will the data survive if a piece of the floor is erased? To answer this, the authors rely on two rules established by previous scientists (Li and Boyle):

Local Indistinguishability: No matter which specific tiling you look at, every small patch of tiles appears with the exact same frequency. If you look at a tiny window, you can't tell which version of the infinite floor you are on.
Local Recoverability: If you erase a patch, the tiles around it contain enough information to perfectly reconstruct what was lost.

The Hat: A Quantum Vault with a Hidden Switch

The authors prove that the Hat tile passes both tests. If you erase a bounded region (a finite hole) from a Hat tiling, the surrounding tiles uniquely determine what should be there. This means the quantum information is safe.

But here is the surprise: The Hat tiling splits into two distinct families (or "sectors").

Family A: Tiles where the "right-handed" Hat is the majority.
Family B: Tiles where the "left-handed" Hat (the mirror image) is the majority.

In the Hat's world, the ratio of right-handed to left-handed tiles is always roughly 6.85 to 1 (specifically $\phi^4 : 1$ , where $\phi$ is the golden ratio). You can't have a 50/50 split. This creates a "superselected" bit—a classical switch that is robust and readable. Even if you erase a chunk of the floor, you can look at the remaining tiles, count the left vs. right hats, and know which family you are in. It's like looking at a crowd and knowing, "This is a crowd of mostly right-handed people," even if you can't see everyone.

The paper proves that this "handedness" bit survives even if you rotate or move the tiling, because the Hat's two families are only swapped by reflections (mirroring), not by simple rotations. Since the laws of physics usually treat rotation as a normal change but don't "gauge away" (ignore) mirror images, this bit remains a permanent, readable label. It's a hybrid memory: it stores quantum data and a classical "handedness" bit simultaneously.

The Spectre: The Switch That Disappears

The Spectre tile also creates a code that protects quantum data. It, too, splits into two families based on orientation (let's call them "even" and "odd" rotations). However, the Spectre is different. Its two families are swapped by a simple 30-degree rotation.

In the quantum world, if you can swap two states just by rotating the system, those states are considered the same "family" when you account for all possible rotations. The authors show that for the Spectre, the "handedness" bit gets "gauged away." It's like trying to tell the difference between a clock showing 12:00 and a clock showing 12:30, but you are allowed to spin the clock face freely. If you can spin it, the difference vanishes. So, while the Spectre protects quantum data, it does not store that extra classical bit in the same robust way the Hat does.

The "What If" Question: Can a Patch Be Retiled?

There is one tiny, nagging question the paper leaves open. The proof that the Hat code works perfectly relies on the idea that you can't take a finite patch of Hat tiles and rearrange them into a different valid pattern.

The Proof: The authors used a computer to check a massive patch of 2,490 tiles. They verified that this specific patch can be tiled in exactly one way.
The Limit: They checked every possible sub-region within that patch (over 2,000 different shapes) and found no alternative tiling.
The Caveat: They haven't mathematically proved this for every possible patch in the infinite universe, only for the ones they simulated. They suspect it's true (just like with the famous Penrose tiles), but they haven't closed the door 100%. If a "rogue" patch existed that could be retiled in two ways, it would break the code's perfect recovery. But for now, the evidence is overwhelming: no such patch has been found up to a very large scale.

The Verdict

This paper confirms that the new "Hat" and "Spectre" tiles are powerful tools for quantum error correction. They are the first single-shape tiles known to do this.

The Spectre is a pure quantum protector, but its extra "switch" disappears if you rotate the system.
The Hat is a hybrid hero. It protects quantum data and carries a robust, classical "handedness" bit that survives rotation. It is the only one in this family that keeps its secret switch safe from the laws of rotation.

The authors have provided the exact mathematical "fingerprint" (frequencies and eigenvalues) for these codes, showing that nature has supplied a new, elegant way to hide information in the geometry of a floor, where the very shape of the tiles acts as the lock and key.

Technical Summary: Quantum Error-Correcting Codes from Aperiodic Monotiles

Problem Statement
The paper addresses the construction of quantum error-correcting codes (QECCs) based on aperiodic tilings, extending the framework established by Li and Boyle for the Penrose tiling. The core challenge is to determine whether the recently discovered aperiodic monotiles—the Hat and the Spectre—satisfy the two necessary conditions for erasure correction: (H1) strong local indistinguishability and (H2) local recoverability. Furthermore, the paper investigates the structure of the resulting code spaces, specifically whether the monotiles introduce new "hybrid" quantum-classical memory structures absent in previous examples like Penrose, Ammann–Beenker, or Fibonacci tilings.

Methodology
The authors apply the Li–Boyle framework, which defines codewords as group-averaged superpositions of tilings over isometry orbits. The methodology involves:

Structural Analysis: Leveraging the "unique hierarchy" theorems established in the discovery papers of the Hat [2] and Spectre [3] to prove that every tiling admits a unique decomposition into supertiles.
Frequency Analysis: Utilizing substitution matrices and Perron–Frobenius theory to calculate asymptotic tile frequencies and establish strong local indistinguishability within specific classes of tilings.
Recoverability Proofs:
- For nonsingular tilings, the authors employ the torus parametrization of the underlying cut-and-project schemes (established in [4] for Hat and [5] for Spectre) to prove that the phase of the tiling is uniquely determined by any unbounded region, ensuring local recoverability.
- For singular tilings, the problem is reduced to a geometric question: whether a bounded region that is a union of tiles can be retiled in a second, distinct way.
Computational Verification: The authors perform exhaustive SAT (Boolean satisfiability) enumerations on certified patches (up to 2,490 tiles for the Hat) to verify the uniqueness of retiling for sub-regions, providing empirical evidence against the existence of singular pairs.
Gauge Group Analysis: The study examines the code structure under different gauge groups ($SE(2)$ vs. $E(2)$ ) to determine how isometries (rotations, translations, reflections) affect the distinguishability of tiling classes.

Key Contributions and Results

Extension to Monotiles: The paper proves that the Li–Boyle construction applies to both the Hat and Spectre monotiles.
- Hat: Strong local indistinguishability is proven for all Hat tilings. Local recoverability is proven unconditionally for all nonsingular Hat tilings. The singular case is reduced to a geometric question regarding the uniqueness of retiling a bounded union of hats. Computational verification confirms no counterexamples exist up to a scale of 2,490 tiles.
- Spectre: Similar results are established for the Spectre, with recoverability proven for nonsingular tilings within each local-indistinguishability (LI) class.
Discovery of Hybrid Quantum-Classical Memory:
- Unlike Penrose, Ammann–Beenker, and Fibonacci tilings, which form a single LI class, both the Hat and Spectre form two distinct LI classes.
- The Hat: The two classes correspond to chirality (majority of "hats" vs. majority of "anti-hats"). These classes are exchanged only by reflections (improper isometries). Under the physically natural gauge group $SE(2)$ (which gauges position and orientation but not parity), the two classes remain distinct. This results in a hybrid memory: a superselected classical bit (the global handedness of the long-range order) coexisting with protected quantum sectors. The bit is readable in any bounded window via the reflected-tile fraction.
- The Spectre: The two classes correspond to rotational subclasses (even vs. odd multiples of $30^\circ$ ). These are exchanged by a $30^\circ$ rotation (a proper isometry). Under $SE(2)$, this rotation is part of the gauge group, causing the two sectors to merge. Consequently, the Spectre does not retain a superselected classical bit under $SE(2)$, unlike the Hat.
Exact Code Parameters:
- The paper provides exact Perron–Frobenius data for both codes.
- Hat: Reflected-hat frequencies are $(3 \mp \sqrt{5})/6$ per class. The separation between sectors is $\Delta = |K|_{\text{tiles}} \sqrt{5}/3$ .
- Spectre: Orientation-class frequencies are $(5 \pm \sqrt{15})/10$ per class. The separation is $\Delta = |K|_{\text{tiles}} \sqrt{15}/5$ (relative to the hexagonal subgroup $G_6$ ).
- The correctable radius scales as $r_n = a \phi^{2n}$ for the Hat and $r_n = a (4+\sqrt{15})^n$ for the Spectre (per $\sigma^2$ step).

Significance and Claims
The paper claims that the Hat monotile is the unique member of the known aperiodic tiling family (including Penrose, Ammann–Beenker, and Fibonacci) that supports a frame-independent, superselected classical bit alongside quantum error-correcting sectors. This arises because the Hat's distinguishing feature (chirality) is invariant under proper isometries but distinct under reflections, whereas the Spectre's distinguishing feature (rotational subclass) is not invariant under proper rotations.

The authors emphasize that this structure represents a "hybrid quantum–classical memory," where the classical label (handedness) is robust against erasure and readable locally, while the quantum information is protected within each sector. The paper remains modest regarding the singular case, noting that while computational evidence is strong, the geometric question of whether a bounded Hat region can be retiled uniquely remains formally open, though the authors have verified it up to a substantial scale.

The work also highlights that the Hat is a reflexible monotile (admitting both handednesses) that carries the chirality bit, while the chiral Spectre (admitting only one handedness) does not, because strict chirality removes the mirrored species entirely, leaving no parity to protect.