Single-shot decoding of the surface code

Quantum error correction is a possible solution to reduce the loss of quantum information in the system. By measuring check operators, we can infer and correct the errors in the system without destroying the encoded quantum states. However, since all measurements can be faulty in a quantum computer, usually, we need to repeat the measurement of the check operators at least as many times as the error correction code distance. Many efforts have been devoted to reduce the repeated measurements needed to perform proper error correction, including using flag qubits, adaptive Shor measurements, or the data-syndrome codes. For certain types of codes, it is shown that single-shot error correction is enough to correct the error, where only one round of measurement of check operators is needed.

Toric code is one of the most famous quantum error correction codes, where we can encode two logical qubits in square lattice of physical qubits with the periodic boundary condition. With the typical local checks of toric codes, single-shot error correction is not possible. However, in our recent paper, we show that by changing the check operators we measure, we can achieve single-shot error correction on toric codes. With this new set of single-shot checks, we achieve a sustainable threshold at 5.6% under the phenomenological noise model, which is twice as high as that of toric code with repeated measurement in the same noise model. The cost of this new set of single-shot checks are the use of high-weight stabilizers. We therefore study these checks in a gate-based noise model that creates increased measurement error with the weight of the checks. Here, there is no single-shot threshold behavior. Instead, for each error rate, there is an optimal size of codes. The conventional checks with repetition measurements outperform the single-shot check set we found in this case.

Our work shows that the performance of an error correction code depends not only on the code space, but also on the check operators we choose. This distinction between code and checks is often lost in the physics literature. We hope it can motivate future work to design new error correction codes and find self-correcting memory due to the mapping of checks and terms in Hamiltonian.