How does the presence of coherent noise, as opposed to stochastic noise, affect the logical error rates and threshold conditions in topological quantum error correction codes like the surface code, and what modifications to syndrome extraction protocols could mitigate these effects?

Quantum error correction (QEC) is essential for maintaining quantum coherence in quantum computation. Topological quantum codes, such as the surface code, are among the most promising candidates for fault-tolerant quantum computation due to their ability to localize errors and correct them using only local measurements. Traditional QEC analysis often assumes stochastic noise, where errors occur randomly and independently. However, in realistic quantum devices, coherent noise — which arises from systematic, non-random imperfections like unitary over-rotations — can accumulate and cause logical errors in more structured ways.

Unlike stochastic errors, coherent errors can interfere constructively, leading to higher logical error rates. This poses new challenges for threshold theorems, which define the error rate below which fault-tolerant computation becomes possible. Addressing coherent noise may require modified syndrome extraction protocols, the use of randomized compiling techniques, or incorporating techniques like noise tailoring. Understanding and mitigating the impact of coherent noise on topological codes is crucial for the development of large-scale quantum computers.