Fault Tolerance with Noisy and Slow Measurements and Preparation

It is not so well known that measurement-free quantum error correction protocols can be designed to achieve fault-tolerant quantum computing. Despite their potential advantages in terms of the relaxation of accuracy, speed, and addressing requirements, they have usually been overlooked since they are expected to yield a very bad threshold. We show that this is not the case. We design fault-tolerant circuits for the 9-qubit Bacon-Shor code and find an error threshold for unitary gates and preparation of p_(p,g)thresh=3.76×10^-5 (30% of the best known result for the same code using measurement) while admitting up to 1/3 error rates for measurements and allocating no constraints on measurement speed. We further show that demanding gate error rates sufficiently below the threshold pushes the preparation threshold up to p_(p)thresh=1/3.