Compressing Syndrome Extraction in Space-Time

Fault-tolerant error correction is crucial for the realization of reliable quantum computation. For a stabilizer code defined by a set of l stabilizer generators, a standard measurement scheme consists of measuring each stabilizer generator once per round, over several rounds of syndrome measurement. However, it is known that a fault-tolerant measurement scheme requiring only O(d log d) measurements exists, where d is the distance of the code.

In this work, we develop a framework for constructing fault-tolerant measurement schedules of varying lengths by combining stabilizer generators according to classical codes. Using this framework, we produce explicit measurement schedules sufficient for fault-tolerant error correction of quantum codes using only O(d log l) measurements if the code is LDPC, and O(d log d log l) measurements if the code is produced via concatenating a smaller code with itself O(log d) times. We also numerically examine the performance of our construction on the surface code and provide evidence that, despite the high weight of the stabilizers produced, our construction is sufficient to produce a positive threshold.