⚛️ quantum physics

Holographic quantum codes with trapped ions

This paper reports the experimental implementation of holographic pentagon and heptagon codes using trapped ions, successfully demonstrating the recovery of logical bulk qubits, verification of the Ryu-Takayanagi entanglement area law, and the emergence of correctable errors from transversal gates, thereby advancing the application of holographic quantum codes in quantum information processing.

Original authors: Alex Steiner, Gerard Anglès Munné, Robert Freund, Ivan Pogorelov, Michael Meth, Robert J. Harris, Gavin Brennen, Thomas M. Stace, Thomas Monz, Rainer Blatt, Felix Huber, Martin Ringbauer

Published 2026-07-21

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CC BY 4.0

Original authors: Alex Steiner, Gerard Anglès Munné, Robert Freund, Ivan Pogorelov, Michael Meth, Robert J. Harris, Gavin Brennen, Thomas M. Stace, Thomas Monz, Rainer Blatt, Felix Huber, Martin Ringbauer

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 the universe as a giant, three-dimensional hologram. It sounds like science fiction, but in the world of modern physics, this idea—called "holography"—is a serious theory. It suggests that all the information inside a 3D space (the "bulk") is actually stored on its 2D surface (the "boundary"), much like how a credit card's hologram looks 3D but is just a flat sticker. This concept bridges the gap between gravity, the stuff that holds planets together, and quantum mechanics, the weird rules that govern tiny particles. Scientists care about this because it might be the missing key to understanding how the universe works at its deepest level. But there's a catch: these ideas are usually just math on a chalkboard. To really know if they work, we need to build them. That's where quantum error correction comes in. Think of it like a game of "telephone" played with a thousand people. If one person whispers the wrong word, the message gets ruined. Quantum computers are super sensitive; a tiny bit of noise can scramble their data. Error correction is the trick of spreading that information out so that even if some parts get messed up, you can still figure out what the original message was.

Now, picture a team of scientists who decided to stop just drawing these holographic maps and actually build one. They used a quantum computer made of trapped ions—tiny, charged atoms floating in a vacuum, held in place by lasers like marbles in an invisible magnetic bowl. Their goal was to test a specific type of holographic code called the "pentagon code" and a cousin called the "heptagon code." These codes are like magical puzzles where information is hidden in the center of a shape, but you can pull it back out just by looking at the edges.

The team successfully built these codes using 12 of their floating atoms. They proved that the "holographic" magic works in real life. For the pentagon code, they showed that you can recover information from the "bulk" (the center) by only looking at a small, nearby section of the "boundary" (the edge). It's as if you could read the entire story of a book just by reading a few sentences on the back cover, without needing the whole book. They even tested a famous rule called the Ryu-Takayanagi formula, which predicts how much "entanglement" (a spooky quantum connection) exists between different parts of the system. Their measurements matched the theory, showing that the geometry of their code really does dictate how information is shared.

They also tackled a trickier version, the heptagon code, which is built from smaller "Steane codes." Here, they tried to perform a specific logic gate (a quantum operation) called the Hadamard gate. They found that doing this operation on the physical atoms didn't work perfectly; it introduced a small, correctable error. But, by using a clever post-processing trick (like a software patch), they fixed the mistake and successfully performed the logical operation. This is a big deal because it shows that these holographic codes aren't just pretty pictures; they can actually process information and handle errors.

The researchers didn't just stop at building it; they tested how robust it was. They intentionally messed up single atoms to see if the code could spot the error. It did. They also tested what happens if you lose a piece of the boundary. They found that if you lose a part of the edge that isn't needed for a specific piece of information, that information is still safe. But if you lose a piece that is needed, that specific part of the message is gone. This confirms the "partial recovery" feature: you don't need the whole system to get the data; you just need the right slice of it.

In short, this paper is a proof-of-concept that holographic quantum codes can be built and tested in a real lab. They demonstrated that these codes can store information in a way that links the inside and outside of a system, allowing for partial recovery and error detection. While the current version is a small-scale experiment with 12 ions, it paves the way for using these holographic ideas to protect future quantum computers from the chaos of the real world. It's a first step in turning a wild theory about the nature of the universe into a practical tool for computing.

Technical Summary: Holographic Quantum Codes with Trapped Ions

Problem Statement
Holography, a concept linking the interior (bulk) of a system to its lower-dimensional boundary, has become central to quantum gravity, condensed matter theory, and quantum information. While theoretical models like the holographic pentagon code (HaPPY code) and heptagon code propose that tensor networks of quantum stabilizer codes can realize key holographic features—such as bulk reconstruction and the Ryu-Takayanagi (RT) entanglement area law—these codes had not yet been implemented in a laboratory setting. Unlike traditional topological codes (e.g., surface codes) which focus on full recovery up to a specific distance, holographic codes are distinguished by their ability to perform partial recovery: reconstructing logical information residing in a specific bulk region using only a nearby subset of boundary qubits. The challenge addressed in this work is the experimental realization of these codes to investigate their error correction capabilities and their connection to holographic physics.

Methodology
The authors experimentally implemented instances of the holographic pentagon and heptagon codes on a trapped-ion quantum processor containing 16 $^{40}\text{Ca}^+$ ions. The logical qubits were encoded into the six ions at the ends of the chain to minimize cross-talk and improve gate fidelity, while four central ions served as idlers.

Code Formulation: The researchers utilized the framework of stabilizer graph codes, where logical states are described as graph states. This approach allowed for efficient preparation, logical operations, and decoding.
- Pentagon Code: Implemented as a minimal instance consisting of 12 physical boundary qubits encoding 4 logical bulk qubits. The state was prepared by contracting four six-qubit absolutely maximally entangled (AME) states, mapped to a graph state $|G_{0000}\rangle$ requiring 28 CZ gates.
- Heptagon Code: Implemented as a $[[12, 2, 3]]$ code constructed by contracting two $[[7, 1, 3]]$ Steane codes. This code belongs to the Calderbank-Shor-Steane (CSS) class, allowing simultaneous readout of X and Z stabilizers.
Experimental Procedures:
- State Preparation and Verification: Fidelity was estimated via Direct Fidelity Estimation (DFE) and by measuring stabilizer generators and logical operators.
- Error Correction: The team demonstrated single-qubit error detection by applying artificial X, Y, or Z errors and measuring syndrome outcomes.
- Partial Decoding (Bulk Reconstruction): The core holographic feature was tested by partitioning the boundary into regions ( $\partial E$ and $\partial F$ ) and applying unitary decoding circuits to recover specific bulk logical qubits from their nearby boundary subsets.
- Logical Operations: Transversal gates (Hadamard, CZ) were implemented. For the heptagon code, the authors specifically investigated the transversal Hadamard gate, which induces a correctable single-qubit error on neighboring logical qubits.
- Entropy Measurement: Von Neumann entropy was measured on boundary regions to verify the Ryu-Takayanagi formula, relating entropy to the length of the bulk cut.

Key Contributions and Results

Experimental Realization: This work presents the first laboratory implementation of holographic quantum codes, specifically the pentagon and heptagon variants, using trapped ions.
Pentagon Code Performance:
- State Preparation: The logical zero state $|G_{0000}\rangle$ was prepared with a measured fidelity of $0.309(13)$ , well above the threshold for a maximally mixed state.
- Error Detection: The code successfully detected single-qubit errors, with each error type producing a unique syndrome.
- Partial Recovery: The team successfully decoded 2, 3, and 4 logical qubits from their respective minimal boundary subsets (cuts 1, 2, and 3). The experimental expectation values matched numerical simulations using a depolarizing noise model.
- Robustness: The system demonstrated that logical qubits could be recovered even if errors occurred on boundary qubits outside the specific decoding region. Conversely, errors on qubits required for a specific decoding cut prevented the recovery of the associated logical information, confirming the locality of the reconstruction.
- Entanglement Area Law: The measured von Neumann entropy for the boundary regions corresponded to the ideal values predicted by the RT formula (proportional to the number of contracted indices in the cut), validating the holographic entanglement structure despite experimental imperfections.
Heptagon Code Performance:
- Logical Operations: The authors demonstrated the implementation of a logical Hadamard gate. They showed that applying a physical Hadamard to all qubits of a logical block induces a transversal logical Hadamard up to a single-qubit error on the neighboring logical qubit. This error was detected via parity measurements on ancilla qubits and corrected via Pauli-frame update in post-processing.
- Efficiency: The CSS structure of the heptagon code allowed for more efficient use of mid-circuit resources compared to the pentagon code.

Significance and Claims
The paper claims that this implementation paves the way for using holographic quantum codes in quantum information processing. By experimentally demonstrating bulk reconstruction and the RT formula, the work provides a tangible link between quantum information theory and the physics of holography. The authors emphasize that holographic codes offer a distinct advantage over traditional codes: the ability to perform partial recovery of information from subregions, which is relevant for scenarios involving "burst errors" (simultaneous errors on neighboring qubits) or when parts of a system are inaccessible.

The authors modestly note that while current constructions require high-weight syndrome measurements, their work serves as a key step toward understanding and simulating bulk-boundary duality. They conclude that these codes provide a natural testbed for exploring the geometric properties of many-body entangled states and their associated error correction capabilities, moving beyond the topological focus of previous experimental efforts.

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