Beyond Single-Shot Fault-Tolerant Quantum Error Correction

Extensive quantum error correction is necessary in order to perform a useful computation on a noisy quantum computer. Moreover, quantum error correction must be implemented based on imperfect parity check measurements that may return incorrect outcomes or inject additional faults into the qubits. To achieve fault-tolerant error correction, Shor proposed to repeat the sequence of parity check measurements until the same outcome is observed sufficiently many times. Then, one can use this information to perform error correction. A basic implementation of this fault tolerance strategy requires
$\Omega (r d^{2})$
parity check measurements for a distance-
$d$
code defined by
$r$
parity checks. For some specific highly structured quantum codes, Bombin has shown that single-shot fault-tolerant quantum error correction is possible using only
$r$
measurements. In this work, we consider a phenomenological noise model for parity check measurements assuming that each bit of a codeword and the measurement outcome suffer from independent bit flips with some error rate
$p$
. For this model, we demonstrate that fault-tolerant quantum error correction can be achieved using
$O(d \log (d))$
measurements for any code with distance
$d \geq \Omega (n^\alpha)$
for some constant
$\alpha > 0$
. Moreover, we prove the existence of a sub-single-shot fault-tolerant quantum error correction scheme using fewer than
$r$
measurements. In some cases, the number of parity check measurements required for fault-tolerant quantum error correction is exponentially smaller than the number of parity checks defining the code. The short measurement sequences constructed generally have high weight and our phenomenological noise model is not realistic in this regime. Our error correction strategy could find applications to small codes and LDPC codes if one can manage to keep the weight of the measured parity checks low.