Beyond Single-Shot Fault-Tolerant Quantum Error Correction

Extensive quantum error correction is necessary in order to perform a useful computation on noisy computer. Moreover, must be implemented based imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant correction, Shor proposed repeat sequence of until same outcome observed sufficiently many times. Then, one can use this information correction. A basic implementation fault tolerance strategy requires $\Omega(r d^2)$ for distance-d code defined by r checks. For some specific highly structured codes, Bombin has shown single-shot possible using only measurements. In work, we demonstrate achieved $O(d \log(d))$ any with distance $d \geq \Omega(n^\alpha)$ constant $\alpha > 0$. prove existence sub-single-shot scheme fewer than cases, number required exponentially smaller checks defining code.