Topological quantum compilation using mixed-integer programming

Quantum computation offers the potential to solve certain problems with substantial speed advantages over classical computing, but its practical realization is hindered by significant challenges associated with error propagation and decoherence. Topological quantum computing addresses these issues by encoding qubits in particular two-dimensional topological condensed matter systems, effectively described by topological quantum field theories, where quantum gates are implemented through sequences of braids of quasiparticles with exotic fractional statistics. The explicit construction of these sequences is a non-trivial task. In this work, we introduce a general optimization framework for a broad class of quantum compilation problems, including the compilation of quantum gates in topological quantum computing. The framework is based on Mixed-Integer Quadratically Constrained Quadratic Programming (MIQCQP), which employs convex relaxations to tackle the underlying non-convex optimization problem. We demonstrate the effectiveness of this framework by compiling a quantum gate from the local equivalence class of the controlled-NOT (CNOT) operation for a non-semisimple topological field theory for the first time. Our results provide a constructive pathway from abstract universality proofs to explicit gate implementations, as a practical tool in topological quantum compilation.