This extreme fragility might make quantum computing sound hopeless. But in 1995, applied mathematician Peter Shor discovered an ingenious way to store quantum information. His coding has two key properties. First, it can tolerate errors that affect only a single qubit. Second, it comes with a program to correct errors as they occur, preventing errors from piling up and derailing calculations. Shore’s discovery is the first example of a quantum error-correcting code, and its two key properties are the defining characteristics of all such codes.
The first property stems from a simple principle: secret information is less vulnerable after being segmented. Spy networks employ similar tactics. Each spy knows very little about the entire network, so even if someone is caught, the organization remains safe. But quantum error-correcting codes take this logic to the extreme. In a quantum spy network, no individual spy would know anything, but together they would know a lot.
Each quantum error-correcting code is a specific recipe for spreading quantum information across many qubits in a collective superposition state. The process effectively converts a cluster of physical qubits into a single virtual qubit. Repeat this process many times with a large number of qubits, and you will have many virtual qubits that can be used to perform calculations.
The physical qubits that make up each virtual qubit are like those unwitting quantum spies. Measuring any of them, you never learn anything about the state of the virtual qubit it belongs to—a property called local indistinguishability. Since each physical qubit encodes no information, errors in a single qubit cannot disrupt calculations. Important information is ubiquitous to some degree, but has no specific location.
“You can’t boil it down to any single qubit,” Cubitt said.
All quantum error-correcting codes can absorb at least one error without any impact on the encoded information, but as errors accumulate, they eventually succumb. This is where the second property of quantum error-correcting codes comes into play—actual error correction. This is closely related to local indistinguishability: since errors in a single qubit do not destroy any information, any error can always be reversed using established procedures specific to each code.
go for a ride
Li Zhi is a postdoctoral fellow at the Perimeter Institute of Theoretical Physics in Waterloo, Canada, and is proficient in quantum error correction theory. But when he spoke to colleague Latham Boyle, the topic escaped him. It was the fall of 2022, and two physicists were taking the evening bus from Waterloo to Toronto. Boyle, an aperiodic tile expert who lived in Toronto at the time and is now at the University of Edinburgh, is a familiar face on those shuttles stuck in traffic.
“Typically, they can be very painful,” Boyle said. “This is like the greatest ever.”
Before that fateful night, Lee and Boyle were aware of each other’s work, but their research areas did not directly overlap, and they had never spoken one-on-one. But like countless researchers in unrelated fields, Li was curious about aperiodic tiling. “It’s hard not to be interested,” he said.