  The traditional method for computation in either the surface code or in the
Raussendorf model is the creation of holes or "defects" within the encoded
lattice of qubits that are manipulated via topological braiding to enact logic
gates. However, this is not the only way to achieve universal, fault-tolerant
computation. In this work, we focus on the Lattice Surgery representation,
which realizes transversal logic operations without destroying the intrinsic 2D
nearest-neighbor properties of the braid-based surface code and achieves
universality without defects and braid based logic. For both techniques there
are open questions regarding the compilation and resource optimization of
quantum circuits. Optimization in braid-based logic is proving to be difficult
and the classical complexity associated with this problem has yet to be
determined. In the context of lattice-surgery-based logic, we can introduce an
optimality condition, which corresponds to a circuit with the lowest resource
requirements in terms of physical qubits and computational time, and prove that
the complexity of optimizing a quantum circuit in the lattice surgery model is
NP-hard.
