Large-scale quantum information processing requires the use of quantum error correcting codes to mitigate the effects of noise in quantum devices. Topological error-correcting codes, such as surface codes, are promising candidates as they can be implemented using only local interactions in a two-dimensional array of physical qubits. Procedures such as defect braiding and lattice surgery can then be used to realize a fault-tolerant universal set of gates on the logical space of such topological codes. However, error correction also introduces a significant overhead in computation time, the number of physical qubits, and the number of physical gates. While optimizing fault-tolerant circuits to minimize this overhead is critical, the computational complexity of such optimization problems remains unknown. This ambiguity leaves room for doubt surrounding the most effective methods for compiling fault-tolerant circuits for a large-scale quantum computer. In this paper, we show that the optimization of a special subset of braided quantum circuits is NP-hard by a polynomial-time reduction of the optimization problem into a specific problem called Planar Rectilinear 3SAT.

翻译：大规模量子信息处理需要使用量子纠错码来减轻量子设备中噪声的影响。拓扑纠错码（如表面码）是很有前景的候选方案，因为它们只需在二维物理量子比特阵列中利用局域相互作用即可实现。随后，缺陷编织和晶格手术等过程可用于在拓扑码的逻辑空间上实现容错通用量子门集。然而，纠错也会引入计算时间、物理量子比特数量和物理门数量的显著开销。尽管优化容错电路以最小化这一开销至关重要，但此类优化问题的计算复杂性仍属未知。这种不确定性使得大规模量子计算机容错电路编译最有效方法的可信度受到质疑。在本文中，我们通过将优化问题多项式归约到特定问题（即平面直线3SAT），证明了一类特殊辫式量子电路子集的优化是NP难的。