Quantum error-correcting codes are essential to the development and scalability of useful quantum computers, because they would allow large-scale computations even in the presence of faulty hardware.
They work through redundancy: a large number of noisy physical qubits are used to implement a smaller number of noiseless logical qubits.
One promising code, the surface code, is the basis of most current quantum computing architectures.
It requires interactions between only neighbouring qubits in a plane and corrects errors optimally given that constraint. Although the constraint of two-dimensional locality is convenient for solid-state qubit architectures based on chips, very large overheads are required to implement noiseless computation. As many as 5,000 noisy physical qubits could be required to implement one noiseless logical qubit. Reducing this overhead requires a more highly connected architecture, but much less is known about optimal performance in higher dimensions or for codes with non-local coupling. For example, it was not known how to design simple yet optimal codes in three spatial dimensions that are analogous to the surface code.
In 2023, Associate Investigator Dominic Williamson and PhD student Nouédyn Baspin developed a construction that generates many examples of such optimal codes. This construction takes any example of a so-called Calderbank–Shor–Steane code as input and outputs a code that is local in three-dimensional space, with the same number of encoded qubits and an improved ability to tolerate errors1. When the input has nice properties—for example, if it is a family of good low-density parity check (LDPC) codes—the output is a three-dimensional topological code with optimal error-correction properties given the locality constraint.
This work provides a practical code construction with optimal error-correction properties, but the study of these new layer codes has only just begun. There are many directions for future work, to investigate the possibility that these layer codes could have practical applications. One direction is to develop an efficient decoder that would allow errors to be corrected in real time, as would be required in a working computer. A second is to investigate whether there are fault-tolerant logic gates that can be performed on these error-correcting codes, allowing them to be used for non-trivial computations and not just as quantum memories.
Looking further ahead, although quantum error correction is an active feedback process in which continual measurements are required to determine where errors have occurred, a grand challenge in the field is to develop a self-correcting memory, for which error correction is a passive process that will occur as long as the memory is sufficiently cold.
Layer codes based on good LDPC codes have an energy barrier that grows with system size, which means they are very promising candidates for self-correcting memories.
This story was first published in the 2023 EQUS annual report, and was written by Kristen Harley.