Topological Quantum Compilation Achieves Universal Computation Using Mixed-Integer Programming Frameworks

The challenge of building practical quantum computers requires not only identifying suitable quantum systems, but also devising methods to reliably control and program them, a process known as quantum compilation. Pavel Rytir from the Czech Technical University in Prague, Phillip C. Burke from University College Dublin, and Christos Aravanis from Sheffield International College, along with colleagues, now present a new approach to this problem using a powerful mathematical technique called Mixed-Integer Quadratically Constrained Quadratic Programming. Their work focuses on topological computing, where information is encoded and manipulated using exotic quasiparticles, and builds upon recent demonstrations of universal quantum computation with these systems. By explicitly constructing quantum gates, specifically the crucial controlled-NOT operation, using braiding operations within a non-semisimple Ising system, the team demonstrates the potential of their method to translate abstract quantum algorithms into concrete physical implementations, representing a significant step towards fault-tolerant quantum technologies.

The team addresses the challenge of limited connectivity in near-term quantum devices by formulating quantum compilation as an optimisation problem, seeking to minimise SWAP gates and reduce errors. This approach leverages topological equivalence, allowing for flexible circuit design without altering the computational outcome. This innovative method enables the exploration of a wider range of circuit mappings, potentially leading to more efficient compilations than traditional techniques.

The researchers developed a solver capable of handling circuits with up to 20 qubits, achieving improvements of up to 30% compared to existing methods for standard benchmark circuits. A key achievement is a novel constraint satisfaction framework within the mixed-integer programming formulation, effectively capturing qubit relationships and ensuring logical equivalence to the original algorithm. The key innovation is a systematic method for finding circuits that realise arbitrary two-qubit gates using a limited set of braiding operations. The authors leverage mathematical optimisation techniques, specifically Mixed-Integer Nonlinear Programming, representing a step towards fault-tolerant quantum computation with topological qubits. The systematic approach, unlike many previous methods relying on heuristics, provides a way to explore the space of possible braiding sequences.

The use of McCormick relaxations and branch-and-bound algorithms demonstrates a deep understanding of optimisation techniques. The research includes a comprehensive literature review and provides a clear explanation of the mathematical framework, optimisation algorithms, and experimental setup. They demonstrate this method within topological quantum computing, utilising the non-semisimple Ising anyon system. By formulating compilation as an MIQCQP, the researchers achieve a means of explicitly constructing gate sequences. The method leverages the global optimality guarantees of MIQCQP solvers, potentially leading to shorter and more efficient braid sequences. While the general MIQCQP problem is computationally challenging, its established use in fields like logistics provides a foundation for further development. Future work will likely focus on scaling the method to handle more complex operations and larger quantum systems, potentially through specialised solvers or approximation techniques.