On fault-tolerance with noisy and slow measurements

It is not so well-known that measurement-free quantum error correction protocols can be designed to achieve fault-tolerant quantum computing. Despite the potential advantages of using such protocols in terms of the relaxation of accuracy, speed and addressing requirements on the measurement process, they have usually been overlooked because they are expected to yield a very bad threshold as compared to error correction protocols which use measurements. Here we show that this is not the case. We design fault-tolerant circuits for the 9 qubit Bacon-Shor code and find a threshold for gates and preparation of $p_{(p,g) thresh}=3.76 \times 10^{-5}$ (30% of the best known result for the same code using measurement based error correction) 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 one can improve the preparation threshold to $p_{(p)thresh} = 1/3$. We also show how these techniques can be adapted to other Calderbank-Shor-Steane codes.