Aperiodic Tilings: A Novel Approach to Quantum Error Correction
The Fascinating World of Aperiodic Tilings
For more than 50 years, aperiodic tilings have captivated mathematicians, enthusiasts, and researchers across various disciplines. Now, two physicists have uncovered a link between aperiodic tilings and a seemingly unconnected field of computer science: the study of how future quantum computers can encode information to safeguard it from errors.
Shor’s Groundbreaking Discovery
In 1995, Peter Shor, a mathematician at MIT, devised an ingenious method for storing quantum information. His encoding possessed two crucial characteristics:
- It could withstand errors that only impacted individual qubits.
- It included a mechanism for correcting errors as they happened, preventing them from accumulating and derailing a computation.
Shor’s finding was the first instance of a quantum error-correcting code, and its two essential properties define all such codes.
The Quantum Spy Network Analogy
Quantum error-correcting codes take the principle of dividing secret information to minimize vulnerability to an extreme. In a quantum spy network, no single spy would possess any knowledge, yet collectively, they would know a great deal.
Each quantum error-correcting code is a unique recipe for spreading quantum information across numerous qubits in a collective superposition state. This process effectively transforms a cluster of physical qubits into a single virtual qubit. By repeating this procedure with a large array of qubits, you can obtain many virtual qubits for performing computations.
Local Indistinguishability and Error Correction
The physical qubits that constitute each virtual qubit are like those oblivious quantum spies. Measuring any one of them reveals nothing about the state of the virtual qubit it belongs to—a property known as local indistinguishability. Since each physical qubit encodes no information, errors in single qubits won’t compromise a computation. The relevant information is somehow everywhere, yet nowhere specific.
“You can’t pin it down to any individual qubit,” Cubitt said.
All quantum error-correcting codes can tolerate at least one error without affecting the encoded information, but they will all eventually succumb as errors accumulate. That’s where the second property of quantum error-correcting codes comes into play—the actual error correction. This is closely tied to local indistinguishability: Because errors in individual qubits don’t destroy any information, it’s always possible to restore the original state.
Bringing Tiling-Based Codes Down to Earth
Li and Boyle have already taken a step towards making tiling-based codes more practical by constructing two other codes in which the underlying quantum system is finite in one case and discrete in the other. The discrete code can also be made finite, but other challenges persist. Both finite codes can only correct errors that are clustered together, whereas the most widely used quantum error-correcting codes can handle randomly distributed errors. It remains unclear whether this is an inherent limitation of tiling-based codes or if it could be overcome with a more sophisticated design.
“There’s lots of follow-up work that can be done,” said Felix Flicker, a physicist at the University of Bristol. “All good papers should do that.”
Exploring Connections and Future Directions
The new discovery raises fundamental questions beyond the technical details. One obvious next step is to determine which other tilings also function as codes. In 2022, mathematicians discovered a family of aperiodic tilings that each only use a single tile. Penrose expressed his fascination with the potential connection between these recent developments and quantum error correction.
Another direction involves investigating connections between quantum error-correcting codes and certain models of quantum gravity. In a 2020 paper, Boyle, Flicker, and the late Madeline Dickens demonstrated that aperiodic tilings appear in the space-time geometry of those models. However, that connection originated from a property of the tilings that plays no role in Li and Boyle’s work. It appears that quantum gravity, quantum error correction, and aperiodic tilings are different pieces of a puzzle whose contours researchers are just starting to grasp. As with aperiodic tilings themselves, understanding how those pieces fit together can be remarkably subtle.
“There are deep roots connecting these different things,” Flicker said. “This tantalizing set of connections is begging to be worked out.”
Original story reprinted with permission from Quanta Magazine, an editorially independent publication of the Simons Foundation whose mission is to enhance public understanding of science by covering research developments and trends in mathematics and the physical and life sciences.
2 Comments
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