Designing fault-tolerant circuits using detector error models

Quantum error-correcting codes, such as subspace, subsystem, and Floquet codes, are typically constructed within the stabilizer formalism, which does not fully capture the idea of fault-tolerance needed for practical quantum computing applications. In this work, we explore the remarkably powerful formalism of detector error models, which fully captures fault-tolerance at the circuit level. We introduce the detector error model formalism in a pedagogical manner and provide several examples. Additionally, we apply the formalism to three different levels of abstraction in the engineering cycle of fault-tolerant circuit designs: finding robust syndrome extraction circuits, identifying efficient measurement schedules, and constructing fault-tolerant procedures. We enhance the surface code's resistance to measurement errors, devise short measurement schedules for color codes, and implement a more efficient fault-tolerant method for measuring logical operators.