Low-Depth Flagged Syndrome Extraction for Calderbank-Shor-Steane Codes of Distance 3

In this work, we present a low-depth flagged syndrome extraction strategy for Calderbank-Shor-Steane (CSS) codes of distance 3. We demonstrate that if one stabilizer generator of a distance-3 CSS code measured using the flagged circuit has and only has two distinct qubit supports with certain stabilizers of opposite types, then the number of stabilizers required for the second round of measurements can be reduced. The number of these stabilizers depends on the weight of the measured generator. This novel strategy utilizes past syndrome information and elements from the stabilizer group, enabling the implementation of a flag-FTEC protocol for distance-3 CSS codes with lower-depth syndrome extraction circuits compared to previously flag fault-tolerant protocol if the total weight of these stabilizers is less than that of stabilizer generators. Based on our strategy, we also improve the flag-FTEC protocol for cyclic CSS codes of distance 3, which requires fewer stabilizer measurements. We utilize the
$[[{14,2,3}]]$
CSS code,
$[[{15,1,3}]]$
quantum Reed-Muller code and
$[[{15,7,3}]]$
cyclic CSS code to demonstrate how the strategy operates. Furthermore, we also provide a low-depth flagged syndrome extraction strategy for a
$[[{19,1,5}]]$
CSS-type code.