Topological Quantum Compilation Using Mixed-Integer Programming

TL;DR

This paper presents a novel optimization framework using mixed-integer quadratic programming to explicitly construct quantum gates in topological quantum systems, focusing on braiding operations in non-semisimple Ising anyon models.

Contribution

It introduces a new optimization-based method for topological quantum compilation, enabling explicit gate construction in complex topological systems.

Findings

Successfully constructed the controlled-NOT gate using braiding in Ising anyon systems.

Demonstrated the applicability of mixed-integer programming to topological quantum gate synthesis.

Showed the method's potential for explicit quantum gate design in topological quantum computing.

Abstract

We introduce the Mixed-Integer Quadratically Constrained Quadratic Programming framework for the quantum compilation problem and apply it in the context of topological quantum computing. In this setting, quantum gates are realized by sequences of elementary braids of quasiparticles with exotic fractional statistics in certain two-dimensional topological condensed matter systems, described by effective topological quantum field theories. We specifically focus on a non-semisimple version of topological field theory, which provides a foundation for an extended theory of Ising anyons and which has recently been shown by Iulianelli et al., Nature Communications {\bf 16}, 6408 (2025), to permit universal quantum computation. While the proofs of this pioneering result are existential in nature, the mixed integer programming provides an approach to explicitly construct quantum gates in topological systems. We demonstrate this by focusing specifically on the entangling controlled-NOT operation, and its local equivalence class, using braiding operations in the non-semisimple Ising system. This illustrates the utility of the Mixed-Integer Quadratically Constrained Quadratic Programming for topological quantum compilation.